Convergence of Fourier Series

6 Key excerpts on "Convergence of Fourier Series"

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  eBook - ePub

  Fourier Analysis

  An Introduction

  • Elias M. Stein, Rami Shakarchi(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  3 Convergence of Fourier Series

  The sine and cosine series, by which one can represent an arbitrary function in a given interval, enjoy among other remarkable properties that of being convergent. This property did not escape the great geometer (Fourier) who began, through the introduction of the representation of functions just mentioned, a new career for the applications of analysis; it was stated in the Memoir which contains his first research on heat. But no one so far, to my knowledge, gave a general proof of it . . .

  G. Dirichlet, 1829

  In this chapter, we continue our study of the problem of Convergence of Fourier Series. We approach the problem from two different points of view.

  The first is “global” and concerns the overall behavior of a function f over the entire interval [0, 2π]. The result we have in mind is “mean-square convergence”: if f is integrable on the circle, then

  At the heart of this result is the fundamental notion of “orthogonality”; this idea is expressed in terms of vector spaces with inner products, and their related infinite dimensional variants, the Hilbert spaces. A connected result is the Parseval identity which equates the mean-square “norm” of the function with a corresponding norm of its Fourier coefficients. Orthogonality is a fundamental mathematical notion which has many applications in analysis.

  The second viewpoint is “local” and concerns the behavior of f near a given point. The main question we consider is the problem of pointwise convergence: does the Fourier series of f converge to the value f (θ ) for a given θ ? We first show that this convergence does indeed hold whenever f is differentiable at θ . As a corollary, we obtain the Riemann localization principle, which states that the question of whether or not

  SN

  (f )(θ ) → f (θ ) is completely determined by the behavior of f in an arbitrarily small interval about θ . This is a remarkable result since the Fourier coefficients, hence the Fourier series, of f depend on the values of f

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  Advanced Mathematical Analysis and Engineering Mathematics

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Theorem. If ƒ belongs to L 2 ([−π, π]), then the Fourier series converg es to ƒ in L 2 ([−π, π]), that is, converges to 0 as N goes to infinity. We have already mentioned that if ƒ is continuously differentiable, then is the n th Fourier coefficient of the derivative ƒ ′. It follows, essentially from the Cauchy-Schwarz inequality, that the Fourier series of ƒ is absolutely summable. The sum of this series is a continuous function, equal to ƒ , since the Fourier series converges in the mean to ƒ : Theorem. If , then the Fourier series converges to ƒ uniformly (and hence also pointwise.) This result can be proven easily if ƒ is further assumed to be C 2 , since in that case tends to zero as . More generally, the Fourier series is absolutely summable, thus converges uniformly to ƒ , provided that ƒ satisfies a Hölder condition of order α > ½. In the absolutely summable case, the inequality proves uniform convergence. Many other results concerning the Convergence of Fourier Series are known, ranging from the moderately simple result that the series converges at x if ƒ is differentiable at x , to Lennart Carleson's much more sophisticated result that the Fourier series of an L 2 function actually converges almost everywhere. These theorems, and informal variations of them that don't specifty the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem. Divergence Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T -periodic function need not converge pointwise. In 1922, Andrey Kolmogorov published an article entitled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere.

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  Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica

  • Kuzman Adzievski, Abul Hasan Siddiqi(Authors)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Fourier analysis lies also at the heart of signal and image processing, including audio, speech, images, videos, radio transmissions, seismic data, and so on. Many modern technological advances, including television, music, CDs, DVDs video movies, computer graphics and image processing, are, in one way or another, founded upon the many results of Fourier analysis. Furthermore, a fairly large part of pure mathematics was invented in connection with the development of Fourier series. This re-markable range of applications qualifies Fourier’s discovery as one of the most important in all of mathematics. 1 2 1. FOURIER SERIES 1.1 Fourier Series of Periodic Functions. In this chapter we will develop properly the basic theory of Fourier analysis, and in the following chapter, a number of important extensions. Then we will be in a position for our main task—solving partial differential equations. A Fourier series is an expansion of a periodic function on the real line R in terms of trigonometric functions. It can also be used to give expansions of functions defined on a finite interval in terms of trigonometric functions on that interval. In contrast to Taylor series, which can only be used to represent functions which have many derivatives, Fourier series can be used to represent functions which are continuous as well as those that are discontinuous. The two theories of Fourier series and Taylor series are profoundly different. A power series either converges everywhere, or on an interval centered at some point c , or nowhere except at the point c . On the other hand, a Fourier series can converge on very complicated and strange sets. Second, when a power series converges, it converges to an analytic function, which is infinitely differentiable, and whose derivatives are represented again by power series. On the other hand, Fourier series may converge, not only to periodic continuous functions, but also to a wide variety of discontinuous functions.

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  Mathematical Methods in Physics

  Partial Differential Equations, Fourier Series, and Special Functions

  • Victor Henner, Tatyana Belozerova, Kyle Forinash(Authors)
  • 2009(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  It is known that the numeric series ∞ summationdisplay n =1 1 n 2 converges. The convergence of the series ∞ summationdisplay n =1 ( A 2 n + B 2 n ) 38 1. Fourier Series can be established, for instance, from Bessel’s inequality (defined later in Equation (1.92)) for the Fourier coefficients of f ′ (x). Therefore, ∞ summationdisplay n =1 (|a n | + |b n |) converges also. Next, notice that for all −π ≤ x ≤ π we have |a n cos nx + b n sin nx| ≤ |a n | + |b n |. From here, using Weierstrass’s criterion (Section 1.2), the Fourier series of f (x) converges uniformly on [−π,π ]. square Very often the problem of term-by-term differentiation and integration of Fourier series arises. It is easy to give examples directly showing that such a differentiation can lead to incorrect results. For example, the Fourier series of f (x) = x defined on [−π,π ] is ∞ summationdisplay n =1 2 n (−1) n +1 sin nx and converges to x on (−π,π ). The derivative of the given function is f ′ (x) = 1 on the interval (−π,π ), but term-by-term differentiation of the series gives ∞ summationdisplay n =1 2(−1) n +1 cos nx, which does not converge. The above problem does not occur for uniformly converging series because, as we know, they can be differentiated term by term. As an ex- ample, we consider f (x) = x 2 on the interval [−π,π ]. From Theorem 1.4, we conclude that the corresponding Fourier series converges uniformly. The main difference with the previous example is that now the periodicity condition, f (−π ) = f (π ), holds. Reading Exercise. Find the Fourier series of x 2 on [−π,π ], differentiate the series term by term, and verify that the result is f ′ (x) = 2x. Reading Exercise. Repeat the previous reading exercise for f (x) = |x| on [−1, 1]. 1.11. Uniform Convergence of Fourier Series 39 In contrast to differentiation, Fourier series can be integrated term by term without strong restrictions on the convergence of the series.

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  Fourier Series In Orthogonal Polynomials

  • Boris Osilenker(Author)
  • 1999(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 3 Convergence and summability of Fourier series inLl 3.1 Fourier series in an abstract Hilbert space Let an orthonormal system {x n } (n € N) be a given in a Hilbert space 'H. Further, let / e H. The numbers c k = c k (f) = (f,x k ) (*€N) are called the Fourier coefficients of the element f with respect to the given or-thonormal system, while the series oo £>***, c fc = (/,**) (*€N) (1.1) is called the Fourier series of the element f eTi. We form the subspace % n = £({xi, £2, • • • > x n})> the elements of which are thus all the possible linear combinations of the first n elements of the given or-thonormal system. We have Theorem 1.1. The partial sum n s n = Y,c k x k ( n £ N ) (1.2) of the Fourier series (1.1) of the element f €% is the projection of the element f on the subspace H n ' s n = npu n f-121 c* = c fc (/) = (/.**) (*€N) 00 X ] c * x *> c * = (/> x *) ( k € N) 00 £ n s n = ^2c k x k ( n £ N ) n «n = npu n f-121 (1.1) (1.2) fc=l 122 Convergence and Summability of Fourier Series Proof. Since / = Sn + (/ ~ S n ) and s n € ri n , it is enough to show that / — x n ± Hn-But it is clear (f-s ni x k ) = 0 (fc = l,2,...), so that / — s n _L ri n -This proves the theorem. We get from this, on using Theorem II. 1.1. Corollary 1.2. p(/;ftn) = | | / -s n | | . So, s n is an element of the best approximation for / in ri. We construct this element. It is very important fact for applications. Further, since ll/H 2 = K|| 2 +||/-Sn|| 2 and Sn 2 = J2cl we obtain ll/-s«ll 2 =ll/ll 2 -E4 (1-3) fc=l Then we get from (1.3) Corollary 1.3. The following BesseVs inequality holds A : = l If we have the sign of equality in (1.4) f>£(/) = Il/H 2 (1-5) fc=l for a certain / €ri, the closure equation (Parseval's equality) is said to be satisfied for/. Remark. The Theorem of Pythagoras asserts that the square of the length of a vector starting from the zero point on n-dimensional Euclidean space is equal to the sum of the squares of its projections on the coordinate axes.

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  Partial Differential Equations

  A First Course

  • Rustum Choksi(Author)
  • 2022(Publication Date)
  • American Mathematical Society

    (Publisher)

  These obser-vations have profound consequences in the applications of Fourier series (for example, in signal processing) and are known as the Gibbs phenomenon . We conclude with comments related to the first observation wherein we claimed that the convergence at noninteger values of ? was slow. What exactly do we mean by slow and, more generally, by speed of convergence? If a function 𝜙 on (−?, ?) is smooth and periodic, then it can be shown that there exists an ? > 0 such that max ?∈[−?,?] |? ? (?) − 𝜙(?)| ≤ ? −?? . This means the convergence is uniform and the partial sums converge exponentially fast . Check this out for yourself: You will observe that just an “average” number ? of partial sums yields an excellent approximation. On the other hand, consider a piece-wise smooth function whose periodic extension is discontinuous. Then at a point ? away from these discontinuities, we would have pointwise convergence but one can only prove |? ? (?) − 𝜙(?)| ≤ ? ? , for some constant ? depending only on 𝜙 . This is what we would call slow convergence. 11.8. What Is the Relationship Between Fourier Series and the Fourier Transform? We remind the reader that the present chapter was written to be independent from the previous Chapter 6 on the Fourier transform. Here we recall the Fourier transform and ask the natural question: What is the relationship? 486 11. Fourier Series Fourier series allow us to encapsulate/reconstruct the entire structure of a func-tion defined on a finite interval with a countable sequence of numbers (the Fourier coefficients). On the other hand, recall the Fourier transform of an integrable function ?(?) on ℝ ̂ ?(?) ≔ ∫ ∞ −∞ ?(?) ? −𝑖?? ??. The Fourier inversion formula tells us how to reconstruct ( synthesize ) the function ? from its Fourier transform ̂ ? : ?(?) = 1 2𝜋 ∫ ∞ −∞ ̂ ?(?) ? 𝑖?? ??.

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