Does the sequence 0,1,0,1,0,1,... converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence an is Cesàro summable to some a if It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it. To discuss summability of Fourier series, we must replace SN wit ...

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Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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A series is said to converge absolutely if the series of absolute values converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum. The Riemann series theorem says that if a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the an are r ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

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There are 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). It is also known that for any function of any Hölder class and any function of ...

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Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation. Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear. This means that the fundamental object of study in potential theory is a linear space of functions. This observation will prove especially important when we consider function space ...

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Potential theory, Potential theory - Definition and comments, Potential theory - Symmetry, Potential theory - Two dimensions, Potential theory - Local behavior, Potential theory - Inequalities, Potential theory - Spaces of harmonic functions

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Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series converges to ex for all x. See also radius of convergence. Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteen ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

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The simplest case is that of L2. According to the Riesz-Fischer theorem, if f is square-integrable then i.e. SN converges to f in the norm of L2. It is easy to see that the opposite is true too: if the limit above is zero, f must be in See also:

Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s and remained open until finally resolved positively in 1966 by Lennart Carleson. Indeed, Carleson showed that the Fourier expansion of any function in L2 converges almost everywhere. Later on Hunt generalized this to Lp for any p > 1. Despite a number of attempts at si ...

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Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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The sum of an infinite series is a limit of partial sums of infinitely many terms. Such a limit can have a finite value; if it has, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes. The simplest convergent infinite series is perhaps It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

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Euler's integral formula for the harmonic numbers follows from the integral identity which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series which is just the Newton's generalized binomial theorem. The interpolating function is in f ...

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Harmonic number, Harmonic number - Introduction, Harmonic number - Applications, Harmonic number - Generalizations, Harmonic number - Generating functions

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The order of growth of Dirichlet's kernel is logarithmic, i.e. See Big O notation for the notation O(1). It should be noted that the actual value 4 / π2 is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for some constant c we have is quite clear when one examines the graph of Dirichlet's kernel. The integral over the n-th peak is bigger than c/n and therefore ...

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Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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Each characterization requires some justification to show that it makes sense. For instance, when the value of the function is defined by a sequence or series, the convergence of this sequence or series needs to be established. Characterizations of the exponential function - Characterization 1. It can be shown that the sequence is an increasing sequence which is bounded above. Since every bounded, increasing sequence of real numbers converges to a unique real number, this characterization makes ...

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Characterizations of the exponential function, Characterizations of the exponential function - Characterizations, Characterizations of the exponential function - Why each characterization makes sense, Characterizations of the exponential function - Characterization 1, Characterizations of the exponential function - Characterization 2, Characterizations of the exponential function - Characterization 3, Characterizations of the exponential function - Equivalence of the characterizations, Characterizations of the exponential function - Equivalence of characterizations 1 and 2, Characterizations of the exponential function - Equivalence of characterizations 1 and 3

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Consider f an integrable function on the interval [0,2π]. For such an f the Fourier coefficients defined by the formula It is common to describe the connection between f and its Fourier series by The notation here means that the sum represents the function in some sense. In order to investigate this more caref ...

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Convergence of Fourier series, Convergence of Fourier series - Preliminaries, Convergence of Fourier series - Convergence at a given point., Convergence of Fourier series - Norm convergence, Convergence of Fourier series - Convergence almost everywhere, Convergence of Fourier series - Absolute convergence, Convergence of Fourier series - Summability, Convergence of Fourier series - Order of growth, Convergence of Fourier series - Multiple dimensions

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A subset E of the circle is called a set of uniqueness, or a U-set, if any trigonometric expansion which converges to zero outside E is identically zero; that is, such that c(n) = 0 for all n. Otherwise E is a set of multiplicity (sometimes called an M-set or a Menshov set). Analogous definitions apply on the real line, and in higher dimensions. In the latter case one needs to specify the order of summation, e.g. "a set of ...

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Set of uniqueness, Set of uniqueness - Definition, Set of uniqueness - Early research, Set of uniqueness - Transformations, Set of uniqueness - Singular distributions, Set of uniqueness - Complexity of structure

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The three pictures on the right demonstrate the phenomenon for a square wave whose Fourier expansion is More precisely, this is the function f which equals π / 4 between 2nπ and (2n + 1)π and − π / 4 between 2(n + 1)π and 2(n + 2)π for every integer n; thus this square wave has a jump discontinuity of height π / 2 at every inte ...

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Gibbs phenomenon, Gibbs phenomenon - Description, Gibbs phenomenon - Formal mathematical description of the phenomenon, Gibbs phenomenon - The square wave example, Gibbs phenomenon - Publications

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Let be a piecewise continuously differentiable function which is periodic with some period L > 0. Suppose that at some point x0, the left limit and right limit of the function f differ by a non-zero gap a: For each positive integer , let SNf be t ...

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Gibbs phenomenon, Gibbs phenomenon - Description, Gibbs phenomenon - Formal mathematical description of the phenomenon, Gibbs phenomenon - The square wave example, Gibbs phenomenon - Publications

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We now illustrate the above Gibbs phenomenon in the case of the square wave described earlier. In this case the period L is 2π, the discontinuity x0 is at zero, and the jump a is equal to π / 2. For simplicity let us just deal with the case when N is even (the case of odd N is very similar). Then we have Substituting x = 0, we obtain See also:

Gibbs phenomenon, Gibbs phenomenon - Description, Gibbs phenomenon - Formal mathematical description of the phenomenon, Gibbs phenomenon - The square wave example, Gibbs phenomenon - Publications

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Asymptotic series, otherwise asymptotic expansions, are infinite series that do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

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The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on the growth behavior of f(t), whereas the two-sided transform is defined in a range a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence. In the two-sided ca ...

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Laplace transform, Laplace transform - Formal definition, Laplace transform - Region of convergence, Laplace transform - Inverse Laplace transform, Laplace transform - Bilateral Laplace transform, Laplace transform - Laplace transform of a function's derivative, Laplace transform - Applications, Laplace transform - Example #1: Solving a differential equation, Laplace transform - Example #2: Deriving the complex impedance for a capacitor, Laplace transform - Example #3: Finding the transfer function from the impulse response, Laplace transform - Relationship to other transforms, Laplace transform - Fourier transform, Laplace transform - Mellin transform, Laplace transform - Z-transform, Laplace transform - Fundamental relationships, Laplace transform - Properties and theorems, Laplace transform - Table of selected Laplace transforms, Laplace transform - Examples: How to apply the properties and theorems, Laplace transform - Example #1: Method of partial fraction expansion, Laplace transform - Example #2: Mixing sines cosines and exponentials, Laplace transform - Example #3, Laplace transform - Example #4: Phase delay

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Series mathematics - Convergence criteria. The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with ...

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Series mathematics, Series mathematics - Infinite series, Series mathematics - Formal definition, Series mathematics - History of the theory of infinite series, Series mathematics - Convergence criteria, Series mathematics - Uniform convergence, Series mathematics - Semi-convergence, Series mathematics - Fourier series, Series mathematics - Some types of infinite series, Series mathematics - Absolute convergence, Series mathematics - Convergence tests, Series mathematics - Power series, Series mathematics - Generalizations

Read more here: » Series mathematics: Encyclopedia II - Series mathematics - History of the theory of infinite series

First, we describe how Euler originally discovered the result. He was considering the harmonic series He had already used the following "product formula" to show the existence of infinitely many primes. (Here, the product is taken over all primes p; in the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes, unless noted otherwise.) Such infinite products are today called Euler products. The product above ...

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Proof that the sum of the reciprocals of the primes diverges, Proof that the sum of the reciprocals of the primes diverges - The harmonic series, Proof that the sum of the reciprocals of the primes diverges - First proof, Proof that the sum of the reciprocals of the primes diverges - Second proof, Proof that the sum of the reciprocals of the primes diverges - Third proof, Proof that the sum of the reciprocals of the primes diverges - External link

Read more here: » Proof that the sum of the reciprocals of the primes diverges: Encyclopedia II - Proof that the sum of the reciprocals of the primes diverges - The harmonic series

We wish to find a function u(x) which solves the following Sturm-Liouville problem: where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example Observe that if k is any integer, then the function is a solution with eigenvalue λ = −k2. We know that the solutions of a S-L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoi ...

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Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes

Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Example

A character on the infinite cyclic group of integers Z under addition is determined by its value at the generator 1. Thus for any character χ on Z, χ(n)=χ(1)n. Moreover, this formula defines a character for any choice of χ(1) in T. Thus it follows easily that algebraically the dual of Z is isomorphic to the circle group T. The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. It is also easily shown that this is the topology of the circle group inherited from the complex numbers. Hence the dual group of See also:

Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Examples