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Analysis

Analysis

An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. See also analytic and synthesis and the Scientific Method. As such, it can be applied in many different fields of study:
- In philosophy:
- philosophical analysis
- In mathematics:
- mathematical analysis
- real analysis - real numbers
- complex analysis - holomorphic functions
- functional analysis - study of spaces of functions
- non-standard analysis - hyperreal numbers
- harmonic analysis - Fourier series
- numerical analysis
- statistical analysis
- meta-analysis
- analysis of variance (ANOVA)
- time-series analysis
- In computing:
- object-oriented analysis and design
- Structured Analysis such as Yourdon
- lexical analysis
- semantic analysis
- syntax analysis
- competitive analysis
- analysis of algorithms
- computer program analysis
- static code analysis
- In music:
- musical analysis
- Schenkerian analysis
- In psychotherapy:
- psychoanalysis
- transactional analysis
- In cryptography:
- cryptanalysis - the decoding of ciphers
- frequency analysis
- In economics:
- fundamentals analysis
- technical analysis - used in predicting stock market trends
- In language studies:
- discourse analysis
- voice analysis
- In signal processing:
- finite element analysis
- principal components analysis
- independent components analysis
- link quality analysis
- path quality analysis In other fields:
- system analysis - in electrical engineering
- systems analysis - study of large systems
- lithic analysis - study of stone tools, in archaeology
- isotope analysis in archaeology and paleontology
- aura analysis - study of bodily auras and energy fields
- dimensional analysis - in physics and engineering
- neutron activation analysis - in chemistry
- life cycle cost analysis
- chemical analysis
- bowling analysis in the sport of cricket
- protocol analysis in knowledge technology ko:해석학 simple:Analyse

Analytic

Analytic may refer to
- Analytic proposition or analytic philosophy, in philosophy
- Analytic geometry, analytic function, analytic continuation, analytic set in mathematics.
- the use of analytic expressions, or periphrasis, in linguistics

Philosophical analysis

Philosophical analysis is a general term for the techniques used by philosophers. These techniques vary across time and place. This article will examine philosophical techniques.

Arguments

In philosophy, logical arguments are used in a somewhat different way to the way than in everyday language. It does not mean a "fight", but is a presentation of the underlying reasons for accepting a particular point of view. A philosophical argument can be contrasted with an argument or debate in rhetoric; the purpose of a rhetorical argument is to persuade the listener, but the purpose of a philosophical argument is to uncover the truth. Because of this philosophers have been concerned to determine what sort of arguments preserve the truth of their assumptions, so that we can rely on them not to lead us astray. Such arguments, in which the truth of the conclusion follows from the truth of the assumptions, are said to be valid, and are the subject of study in logic.

Meaning

Words are the weapon of choice in doing philosophy. They can be used as rhetorical tools, to make a case seem more persuasive than it might otherwise be, or they can be used to lay out the significance of the various parts of an argument or position so that one can see what is going on more clearly. The hard part is working out which is which.

Logic

External Links

- [http://plato.stanford.edu/entries/analysis/ Article on analysis] at the Stanford Encyclopedia of Philosophy.
- [http://www.swemorph.com/pdf/anaeng-r.pdf Analysis and Synthesis: On Scientific Method based on a study by Bernhard Riemann] From the [http://www.swemorph.com The Swedish Morphological Society] Analysis

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Mathematical analysis

Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis has its beginnings in the rigorous formulation of calculus.

History

Greek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In the 12th century the Indian mathematician Bhaskara gave an example of what would now be called a "differential coefficient" and the basic idea behind what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama expressed various trigonometric functions as infinite series, and estimated the magnitude of the error terms created by truncating these series. In Europe, analysis originated in the 17th century, with the independent invention of calculus by Newton and Leibniz. In the 17th and 18th centuries, analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate discrete problems by continuous ones. All through the 18th century the definition of the concept of function was a subject of debate among mathematicians. In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of the Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Subdivisions

Analysis is nowadays divided into the following subfields:
- Real analysis, the formally rigorous study of derivatives and integrals of real-valued functions. This includes the study of limits, power series and measures.
- Functional analysis studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
- Harmonic analysis deals with Fourier series and their abstractions.
- Complex analysis, the study of functions from the complex plane to the complex plane which are complex differentiable.
- p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
- Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. It is normally classed as model theory. Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with 'straight' analysis is large.
- ar : تحليل رياضي ja:解析学

Real analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. It is a sophisticated theory of the 'numerical function' idea, and contains modern theories of generalized functions. The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. Then the real numbers are either introduced axiomatically, or they are constructed from Cauchy sequences or Dedekind cuts of rational numbers. Initial consequences are derived, most importantly the properties of the absolute value such as the triangle inequality and Bernoulli's inequality. The concept of convergence, central to analysis, is introduced via limits of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power series serve to cleanly define several central functions, such as the exponential function and the trigonometric functions. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next. The concept of continuity may now be defined via limits. One can show that the sum, product, composition and quotient of continuous functions is continuous, excluding at points where the denominator function has value zero, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem. Then one can do integration (Riemann and Lebesgue) and prove the fundamental theorem of calculus, typically using the mean value theorem. At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated. We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.

External links

- [http://www.math.unl.edu/~webnotes/contents/chapters.htm Analysis WebNotes] by John Lindsay Orr
- [http://www.shu.edu/projects/reals/index.html Interactive Real Analysis] by Bert G. Wachsmuth
- [http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html A First Analysis Course] by John O'Connor
- ja:%E5%AE%9F%E8%A7%A3%E6%9E%90

Functional analysis

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. It has its historical roots in the study of transformations, such as the Fourier transform, and in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra.

Normed vector spaces

In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in the mathematical formulation of quantum mechanics. More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm. An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C
- -algebras and other operator algebras.

Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ₀) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper subspace which is invariant. Many special cases have already been proven.

Banach spaces

General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example. For any real number p ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see L^p spaces). In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. This is explained in the dual space article. The notion of derivative is extended to arbitrary functions between Banach spaces. It turns out that the derivative of a function at a certain point is really a continuous linear map.

Major and foundational results

These are important results of functional analysis:
- The uniform boundedness principle is a result on sets of operators with tight bounds.
- One spectral theorem (there are more of them) gives an integral formula for normal operators on a Hilbert space. It is of central importance in the mathematical formulation of quantum mechanics.
- The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion. Another implication is the non-triviality of dual spaces.
- The open mapping theorem and closed graph theorem. See also: List of functional analysis topics.

Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. Many very important theorems require the Hahn-Banach theorem, which itself is a form of the axiom of choice that is strictly weaker than the Boolean prime ideal theorem.

Points of view

Functional analysis as it currently stands includes a number of directions:
- soft analysis, the approach to mathematical analysis based generally on topological groups, topological rings and topological vector spaces;
- geometry of Banach spaces, a combinatorial approach as in the work of Jean Bourgain;
- noncommutative geometry as developed by Alain Connes, based partly on previous ideas such as George Mackey's approach to ergodic theory;
- connection with quantum mechanics, narrowly defined in mathematical physics or broadly interpreted as by Israel Gelfand to include most types of representation theory.

References

- Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980
- Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001
- Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0444517901
- Dunford, N. and Schwartz, J.T. : Linear Operators, General Theory, and other 3 volumes, includes visualization charts
- Brezis, H.: Analyse Fonctionnelle, Dunod
- Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963
- Lebedev, L.P. and Vorovich, I.I.: Functional Analysis in Mechanics, Springer-Verlag, 2002
- ja:関数解析学

HyperReal numbers

In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as
- R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about
- R. An important property of
- R is that it has infinitely large as well as infinitesimal numbers, where an infinitely large number is one that is larger than all numbers representable in the form :

1 + 1 + \cdots + 1.

The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Kanovei and Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis. When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a limit by Cauchy, Weierstrass and others, they were largely abandoned. However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from the use of them in nonstandard analysis, have no necessary relationship to model theory or first order logic.

The transfer principle

Historically, the concept of number has been repeatedly generalized. At each step in this process of generalization, mathematicians knew that they wished to retain as many properties as possible from the earlier concepts of numbers. However, some properties always had to be given up. In the case of the hyperreals, a long historical delay in their development was caused by uncertainty among mathematicians as to exactly which properties could be retained, and which would have to be given up. The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it: ::

\forall x \in \mathbb \quad \exists y \in\mathbb\quad   x < y

The same will then also hold for hyperreals: ::

\forall x \in \star \mathbb \quad \exists y \in\star \mathbb\quad  x < y

Another example is the statement that if you add 1 to a number you get a bigger number: ::

\forall x \in \mathbb \quad x < x+1

which will also hold for hyperreals: ::

\forall x \in \star \mathbb \quad  x < x+1

The correct general statement that formulates these equivalences is called the transfer principle. Note that in many formulas in analysis quantification is over higher order objects such as functions and sets which makes the transfer principle somewhat more subtle than the above examples suggest. The transfer principle however doesn't mean that R and
- R have identical behavior. For instance, in
- R there exists an element w such that ::

1 but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like

w is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals. The hyperreals
- R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology. The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah, in the paper linked to at the end of this article, have found a method that gives an explicit construction, at the cost of a significantly more complicated treatment.)

The ultrapower construction

We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example, :

(a_0, a_1, a_2, \ldots) + (b_0, b_1, b_2, \ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, \ldots)

and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, ...) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say,

7+\epsilon

, where

\epsilon

is a certain infinitesimal number. Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion: :

(a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff  a_0 \leq b_0 \wedge a_1 \leq b_1 \wedge a_2 \leq b_2 \ldots

but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is a only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters which do not contain any finite sets. (The good news is that the axiom of choice guarantees the existence of many such U, and it turns out that it doesn't matter which one we take; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that "matter": We write (a₀, a₁, a₂, ...) ≤ (b₀, b₁, b₂, ...) if and only if the set of natural numbers is in U. This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field
- R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A, and then to define
- R as A/I; as the quotient of a commutative ring by a maximal ideal,
- R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent. The field A/U is an ultrapower of

\Bbb

. Since this field contains R it has cardinality at least the continuum. Since A has cardinality :

(2^)^ = 2^ =2^,\,

it is also no larger than

2^

, and hence has the same cardinality as R. As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R. Thus in some sense of "larger" we do not need to go to a larger field to do nonstandard analysis. One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the continuum hypothesis false we can prove that there are non-order-isomorphic pairs of fields which are both countably indexed ultrapowers of the reals. For more information about this method of construction, check out ultraproducts and ultrapowers.

An intuitive approach to the ultrapower construction

The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt (see the references below). Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense, the true infinitesimals are the classes of sequences that contain a sequence converging to zero. Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply add and subtract them, but not always divide by non-zero. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is,

a_n=0

for all

n

. In our ring of sequences one can get

ab=0

with neither

a=0

nor

b=0

. Thus, if for two sequences

a, b

one has

ab=0

, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As result, the classes of sequences that differ by some sequence declared zero will form a field which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, they will be represented by the sequences converging to infinity). Also every hyperreal which is not infinitely large will be infinitely close to an ordinary real, in other words, it will be an ordinary real + an infinitesimal. This construction is parallel to the construction or the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, let us consider the zero sets of our sequences, that is, the

z(a)=\

, that is,

z(a)

is the set of indexes

i

for which

a_i=0

. It is clear that if

ab=0

, then the union of

z(a)

and

z(b)

is N (the set of all natural numbers), so: :(i) one of the sequences that vanish on 2 complementary sets should be declared zero also : (ii) if

a

is declared zero,

ab

should be declared zero too, no matter what

b

is. and :(iii) if both

a

and

b

are declared zero, then

a^2+b^2

should also be declared zero. Now the idea is to single out a bunch U of subsets X of N and to declare that

a=0

if and only if

z(a)

belongs to U. From the conditions (i), (ii) and (iii) one can see that :(i) From 2 complementary sets one belongs to U :(ii) Any set containing any set that belong to U, belongs to U. :(iii) An intersection of any 2 sets belonging to U belongs to U. Also :(iv) we don't want an empty set to belong to U because then everything becomes zero because every set contains an empty set. Any family of sets that satisfies (ii)-(iv) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (i) holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10) such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers (exercise). Any ultrafilter containing a finite set is trivial (exercise). It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter can be added as an extra axiom, it's weaker than the axiom of choice (that says that for any bunch of nonempty sets there is a function that picks an element from any of them, f(X) is an element of X). Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter, exercise) and do our construction, we get the hyperreal numbers as a result. The infinitesimals can be represented by the non-vanishing sequences converging to zero in the usual sense, that is with respect to the Fréchet filter (exercise). If

f

is a real function of a real variable

x

then f naturally extends to a hyperreal function of a hyperreal variable by composition: :

f(\)=\

where

\

means "the equivalence class of the sequence

\dots

relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter. All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. One can prove that any finite (that is, such that

|x| < a

for some ordinary real

a

) hyperreal

x

will be of the form

y+d

where

y

is an ordinary (called standard) real and

d

is an infinitesimal. It is parallel to the proof of the Bolzano-Weierstrass lemma that says that one can pick a convergent subsequence form any bounded sequence, done by bisection, the property (i) of the ultrafilters is again crucial. Now one can see that

f

is continuous means that

f(a)-f(x)

is infinitely small whenever

x-a

is and

f

is differentiable means that :

(f(x)-f(a))/(x-a)-f'(a)\quad

is infinitely small whenever

x-a

is. Remarkably, if one allows

a

to be hyperreal, the derivative will be automatically continuous (because,

f

being differentiable at

x

, :

f'(x)-(f(x)-f(a))/(x-a)=f'(x)-(f(a)-f(x))/(a-x)\quad

is infinitely small when

x-a

is, therefore

f'(x)-f'(a)\quad

is also infinitely small when

x-a

is).

Infinitesimal and infinite numbers

A hyperreal number r is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number. Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because the ultrafilter U contains all index sets whose complement is finite). A hyperreal number x is called finite (or limited by some authors) if there exists a natural number n such that -n < x < n; otherwise, x is called infinite (or illimited). Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A non-zero number x is infinite if and only if 1/x is infinitesimal. The finite elements of F of
- R form a local ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x. This operation is an order-preserving homomorphism and hence well-behaved both algebraically and order theoretically. However, it is order-preserving but not isotonic, which means

x \le y

implies

\operatorname(x) \le \operatorname(y)

, but it is not the case that

x < y

implies

\operatorname(x) < \operatorname(y)

- We have, if both x and y are finite, ::

\operatorname(x + y) = \operatorname(x) + \operatorname(y)

\operatorname(x y) = \operatorname(x)  \operatorname(y)

- If x is finite and not infinitesimal. ::

\operatorname(1/x)  = 1 /  \operatorname(x)

- x is real if and only if ::

\operatorname(x) = x

The map st is locally constant, which entails that its derivative is identically zero and that it is continuous with respect to the order topology on the finite hyperreals.

Hyperreal fields

Suppose X is a Tychonoff space, also called a T_3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can have the cardinality of the continuum, in which case F is isomorphic as a field to R, but is not order isomorphic to R. An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra

\Bbb^\kappa

of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory. We give a particular example, commonly used in nonstandard analysis, below. Compare with:
- Surreal numbers
- Superreal numbers
- Real closed fields

References

- H. Garth Dales and W. Hugh Woodin: Super-Real Fields, Clarendon Press, 1996.
- L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.
- Robert Goldblatt, Lectures on the hyperreals : an introduction to nonstandard analysis, Springer, 1998.
- Abraham Robinson: Nonstandard Analysis, Princeton University Press 1966. The classic introduction to nonstandard analysis.

External links

- [http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Approach Using Infinitesimals by H. Jerome Keisler] includes an axiomatic treatment of the hyperreals, and is freely available under a Creative Commons license
- Jordi Gutierrez Hermoso: [http://mathforum.org/dr.math/faq/analysis_hyperreals.html Nonstandard Analysis and the Hyperreals]. A gentle introduction.
- Vladimir Kanovei and Saharon Shelah: [http://shelah.logic.at/files/825.pdf A definable nonstandard model of the reals], Journal of Symbolic Logic 69 (2004) pp. 159-164. Category: Model theory Category: Field theory Category: real closed field

Harmonic analysis

Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics", hence the name "harmonic analysis." In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. The classical Fourier transform on Rⁿ is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley-Wiener theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very elementary form of an uncertainty principle in a harmonic analysis setting. See also classic harmonic analysis. Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.

Abstract harmonic analysis

One of the more modern branches of harmonic analysis, having its roots in the mid-twentieth century, is analysis on topological groups. The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on locally compact groups. The theory for abelian locally compact groups is called Pontryagin duality; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes. It is developed in detail on its dedicated page. Harmonic analysis studies the properties of that duality and Fourier transform; and attempts to extend those features to different settings, for instance to the case of non-abelian Lie groups. For general nonabelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. For compact groups, the Peter-Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations. This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure. If the group is neither abelian nor compact, no general satisfactory theory is currently known. By "satisfactory" one would mean at least the equivalent of Plancherel theorem. However, many specific cases have been analyzed, for example SL_n. In this case, it turns out that representations in infinite dimension play a crucial role.

Other branches

- Study of the eigenvalues and eigenvectors of the Laplacian on domains, manifolds and (to a lesser extent), graphs, is also considered a branch of harmonic analysis. See e.g., hearing the shape of a drum.
- Harmonic analysis on Euclidean spaces deals with properties of the Fourier transform on Rⁿ which have no analog on general groups. For example, the fact that the Fourier transform is invariant to rotations. Decomposing the Fourier transform to its radial and spherical components leads to topics such as Bessel functions and spherical harmonics. See the book reference.
- Harmonic analysis on tube domains is concerned with generalizing properties of Hardy spaces to higher dimensions.

References

Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. ISBN 069108078X Yitzhak Katznelson, An introduction to harmonic analysis, Third edition. Cambridge University Press, 2004. ISBN 0-521-83829-0; 0-521-54359-2 Category:Mathematical analysis
-

Fourier series

The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. Intuitively, one can use Fourier series to divide certain large problems into more manageable pieces. More precisely, a Fourier series is a representation of a periodic function with period 2π as a sum of periodic functions of the form :

x\mapsto e^,

which are the harmonics of e^{i x}. By Euler's formula, the series can be expressed equivalently in terms of sine and cosine functions. This can be generalized to periodic functions of any positive period. Fourier was the first to study systematically such infinite series, after preliminary investigations by Euler, d'Alembert, and Daniel Bernoulli. He applied these series to the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822. From a modern point of view, Fourier's results are somewhat informal, due in no small part to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality. Many other Fourier-related transforms have since been defined, extending to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis.

Definition of Fourier series

Suppose that f(x), a complex-valued function of a real variable, is periodic with period 2π, and is square-integrable over the interval from −π to π. Let :

F_n =\frac\int_^\pi f(x)\,e^\,dx.

Each F_n is called a Fourier coefficient. Then, the Fourier series representation of f(x) is given by :

f(x) = \sum_^ F_n \,e^.

Each term in this sum is called a Fourier mode or a harmonic. In the important special case of a real-valued function f(x), one often uses the equality :

e^=\cos(nx)+i\sin(nx) \,\!

(derived from Euler's formula) to equivalently represent f(x) as an infinite linear combination of functions of the form

\cos(nx) \,\!

and

\sin(nx) \,\!

, that is :

f(x) = \fraca_0 + \sum_^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right]

, where :

a_n = \frac\int_^\pi f(x)\cos(nx)\,dx

and

b_n = \frac\int_^\pi f(x)\sin(nx)\,dx

which corresponds to

F_n = (a_n - i b_n) / 2 \,

and

F_ = F_n^ -

, and therefore

F_ = a_0 / 2.\,

Example

Let f(x) = x be the identity function for x from −π to π. Outside this domain, the Fourier series implicitly requires that we define the function periodically. We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions. :

a_n=\frac\int_^f(x)\cos(nx)\,dx= \frac\int_^x \cos(nx)\,dx = 0,

b_n= \frac\int_^f(x)\sin(nx)\,dx=\frac\int_^ x \sin(nx)\, dx

=\frac\int_^ x\sin(nx)\, dx= \frac\left(
\left[-\frac\right]_0^+\left[\frac\right]_0^
\right)=(-1)^\frac.

Notice that a_n are 0 because x and x cos(nx) are odd functions. Hence the Fourier series for f(x) = x is: :

f(x)=x=\frac + \sum_^(a_n\cos(nx)+b_n\sin(nx))

=\sum_^(-1)^\frac \sin(nx), \quad \forall x\in (-\pi,\pi).

For an application of this Fourier series, see the value of the Riemann zeta function at s = 2.

Convergence of Fourier series

While the Fourier coefficients a_n and b_n can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. The simplest answer is that if f is square-integrable then :

\lim_\int_^\pi\left|f(x)-\sum_^
F_n\,e^\right|^2\,dx=0

(this is convergence in the norm of the space L²). There are also many known tests that ensure that the series converges at a given point x. For example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. A discussion of the counterexample, along with other positive and negative results in the general spirit of "for functions of type X, the Fourier series converges in sense Y" may be found in Convergence of Fourier series.

Orthogonality

The Fourier basis functions are orthogonal in the discrete space :

\sum_^\infty e^e^=2\pi\sum_^\infty
\delta(x-y+2\pi n)=2\pi\,\delta_(x-y)

where δ(x) is the Dirac delta function and δ_T(x) is the Dirac comb function. The Fourier basis functions are orthogonal in the continuous space as well: :

\frac\int_^\pi e^e^\,dx = \delta_

where δ_nm is the Kronecker delta function.

Some positive consequences of the homomorphism properties of exp

Because "basis functions" e^ikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities:

Shifting property

If :

g(x)=f(x-y) \,\!

then (if G is the transform of g) :

G_k = e^F_k. \,\!

Convolution theorems

:Main article: Convolution If h(t) is the cyclic convolution of f(t) and g(t): :

h(t)=\int_^\pi f(t')g(t-t')\,dt'

where g(t) = g(t + 2nπ), then the Fourier series transforms are related by: :

H_n=2\pi\,F_nG_n.\,

Conversely, if H_n = 2πF_nG_n, then h(t) will be the cyclic convolution of f(t) and g(t). In the discrete space, if H_n is the discrete convolution of F_n and G_n: :

H_k=\sum_^\infty F_n G_

then the inverse transforms are related by: :

h(t)=f(t)g(t)\,

and conversely, if h(t) = f(t)g(t), then H_n will be the discrete convolute of F_n and G_n. These theorems may be proven using the orthogonality relationships.

Plancherel's and Parseval's theorem

Another important property of the Fourier series is the Plancherel theorem :

\sum_^\infty F_nG^ - _n = \frac \int_^\pi f(x)g^ - (x)\,dx.

Parseval's theorem, a special case of the Plancherel theorem, states that :

\sum_^\infty |F_n|^2 = \frac \int_^\pi |f(x)|^2 \,dx

which can be restated for the real-valued f(x) case above, :

\frac + \frac \sum_^\infty \left( a_n^2 + b_n^2 \right) = \frac \int_^\pi f(x)^2\, dx.

These theorems may be proven using the orthogonality relationships.

General formulation

The useful properties of Fourier series are largely derived from the orthogonality and homomorphism property of the functions

e^ \,\!

. Other sequences of orthogonal functions have similar properties, although some useful identities concerning e.g. convolutions are no longer true once we lose the homomorphism property. Examples include sequences of Bessel functions and orthogonal polynomials. Such sequences are commonly the solutions of a differential equation; a large class of useful sequences are solutions of the so-called Sturm-Liouville problems.

References

- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0486633314

External links

- [http://www.falstad.com/fourier/ Java applet] shows Fourier series expansion of an arbitrary function Category:Fourier analysis
- Category:Mathematical series ko:푸리에 급수 th:อนุกรมฟูริเยร์

Statistics

Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. It is applicable to a wide variety of academic disciplines from the physical and social sciences to the humanities, as well as to business, government, and industry. Once data is collected, either through a formal sampling procedure or by recording responses to treatments in an experimental setting (cf experimental design), or by repeatedly observing a process over time (time series), graphical and numerical summaries may be obtained using descriptive statistics. Patterns in the data are modeled to draw inferences about the larger population, using inferential statistics to account for randomness and uncertainty in the observations. These inferences may take the form of answers to essentially yes/no questions (hypothesis testing), estimates of numerical characteristics (estimation), prediction of future observations, descriptions of association (correlation), or modeling of relationships (regression). The framework described ab