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# Convergence of Fourier series

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 Title: Convergence of Fourier series Author: World Heritage Encyclopedia Language: English Subject: Collection: Fourier Series Publisher: World Heritage Encyclopedia Publication Date:

### 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

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.