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Convergence of Fourier series

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Title: Convergence of Fourier series  
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Convergence of Fourier series

In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.

Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean.

Contents

  • Preliminaries 1
  • Magnitude of Fourier coefficients 2
  • Pointwise convergence 3
  • Uniform convergence 4
  • Absolute convergence 5
  • Norm convergence 6
  • Convergence almost everywhere 7
  • Summability 8
  • Order of growth 9
  • Multiple dimensions 10
  • Notes 11
  • References 12

Preliminaries

Consider ƒ an integrable function on the interval [0,2π]. For such an ƒ the Fourier coefficients \widehat{f}(n) are defined by the formula

\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}f(t)e^{-int}\,dt, \quad n \in \mathbf{Z}.

It is common to describe the connection between ƒ and its Fourier series by

f\sim \sum_n \widehat{f}(n)e^{int}.

The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined:

S_N(f;t)=\sum_{n=-N}^N \widehat{f}(n)e^{int}.

The question here is: Do the functions S_N(f) (which are functions of the variable t we omitted in the notation) converge to ƒ and in which sense? Are there conditions on ƒ ensuring this or that type of convergence? This is the main problem discussed in this article.

Before continuing, the Dirichlet kernel must be introduced. Taking the formula for \widehat{f}(n), inserting it into the formula for S_N and doing some algebra gives that

S_N(f)=f * D_N\,

where ∗ stands for the periodic convolution and D_N is the Dirichlet kernel, which has an explicit formula,

D_n(t)=\frac{\sin((n+\frac{1}{2})t)}{\sin(t/2)}.

The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely

\int |D_n(t)|\,dt \to \infty

a fact that plays a crucial role in the discussion. The norm of Dn in L1(T) coincides with the norm of the convolution operator with Dn, acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional ƒ → (Snƒ)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n → ∞.

Magnitude of Fourier coefficients

In applications, it is often useful to know the size of the Fourier coefficient.

If f is an absolutely continuous function,

\left|\widehat f(n)\right|\le {K \over |n|}

for K a constant that only depends on f.

If f is a bounded variation function,

\left|\widehat f(n)\right|\le =\infty.

It is not known whether this example is best possible. The only bound from the other direction known is log n.

Multiple dimensions

Upon examining the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses. For example, in two dimensions, one may define

S_N(f;t_1,t_2)=\sum_{|n_1|\leq N,|n_2|\leq N}\widehat{f}(n_1,n_2)e^{i(n_1 t_1+n_2 t_2)}

which are known as "square partial sums". Replacing the sum above with

\sum_{n_1^2+n_2^2\leq N^2}

lead to "circular partial sums". The difference between these two definitions is quite notable. For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of \log^2 N while for circular partial sums it is of the order of \sqrt{N}.

Many of the results true for one dimension are wrong or unknown in multiple dimensions. In particular, the equivalent of Carleson's theorem is still open for circular partial sums. Almost everywhere convergence of "square partial sums" (as well as more general polygonal partial sums) in multiple dimensions was established around 1970 by Charles Fefferman.

Notes

  1. ^ Jackson (1930), p21ff.
  2. ^ Stromberg (1981), Exercise 6 (d) on p. 519 and Exercise 7 (c) on p. 520.

References

Textbooks

  • Dunham Jackson "The theory of Approximation", AMS Colloquium Publication Volume XI, New York 1930.
  • Nina K. Bary, A treatise on trigonometric series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book. The Macmillan Co., New York 1964.
  • Antoni Zygmund, Trigonometric series, Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. ISBN 0-521-89053-5
  • Yitzhak Katznelson, An introduction to harmonic analysis, Third edition. Cambridge University Press, Cambridge, 2004. ISBN 0-521-54359-2
  • Karl R. Stromberg, "Introduction to classical analysis", Wadsworth International Group, 1981. ISBN 0-534-98012-0
The Katznelson book is the one using the most modern terminology and style of the three. The original publishing dates are: Zygmund in 1935, Bari in 1961 and Katznelson in 1968. Zygmund's book was greatly expanded in its second publishing in 1959, however.

Articles referred to in the text

  • Paul du Bois-Reymond, Ueber die Fourierschen Reihen, Nachr. Kön. Ges. Wiss. Göttingen 21 (1873), 571–582.
This is the first proof that the Fourier series of a continuous function might diverge. In German
  • Andrey Kolmogorov, Une série de Fourier–Lebesgue divergente presque partout, Fundamenta Mathematicae 4 (1923), 324–328.
  • Andrey Kolmogorov, Une série de Fourier–Lebesgue divergente partout, C. R. Acad. Sci. Paris 183 (1926), 1327–1328
The first is a construction of an integrable function whose Fourier series diverges almost everywhere. The second is a strengthening to divergence everywhere. In French.
  • Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966) 135–157.
  • Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), 235–255. Southern Illinois Univ. Press, Carbondale, Ill.
  • Charles Louis Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551–571.
  • Michael Lacey and Christoph Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7:4 (2000), 361–370.
  • Ole G. Jørsboe and Leif Mejlbro, The Carleson–Hunt theorem on Fourier series. Lecture Notes in Mathematics 911, Springer-Verlag, Berlin-New York, 1982. ISBN 3-540-11198-0
This is the original paper of Carleson, where he proves that the Fourier expansion of any continuous function converges almost everywhere; the paper of Hunt where he generalizes it to L^p spaces; two attempts at simplifying the proof; and a book that gives a self contained exposition of it.
  • Dunham Jackson, Fourier Series and Orthogonal Polynomials, 1963
  • D. J. Newman, A simple proof of Wiener's 1/f theorem, Proc. Amer. Math. Soc. 48 (1975), 264–265.
  • Jean-Pierre Kahane and Yitzhak Katznelson, Sur les ensembles de divergence des séries trigonométriques, Studia Math. 26 (1966), 305–306
In this paper the authors show that for any set of zero measure there exists a continuous function on the circle whose Fourier series diverges on that set. In French.
  • Sergei Vladimirovich Konyagin, On divergence of trigonometric Fourier series everywhere, C. R. Acad. Sci. Paris 329 (1999), 693–697.
  • Jean-Pierre Kahane, Some random series of functions, second edition. Cambridge University Press, 1993. ISBN 0-521-45602-9
The Konyagin paper proves the \sqrt{\log n} divergence result discussed above. A simpler proof that gives only log log n can be found in Kahane's book.
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