Convergence of Fourier series

In mathematics, the question of whether the Fourier series of a given periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics.

are 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.

To investigate this more carefully, the partial sums must be defined: The question of whether a Fourier series converges is: Do the functions

The Dirichlet–Dini Criterion states that:[4] if ƒ is 2π–periodic, locally integrable and satisfies then (Snf)(x0) converges to ℓ.

This implies that for any function f of any Hölder class α > 0, the Fourier series converges everywhere to f(x).

It is also known that for any periodic function of bounded variation, the Fourier series converges.

In general, the most common criteria for pointwise convergence of a periodic function f are as follows: There exist continuous functions whose Fourier series converges pointwise but not uniformly.

[8] However, the Fourier series of a continuous function need not converge pointwise.

Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L1(T) and the Banach–Steinhaus uniform boundedness principle.

As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive.

However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere.

It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by[9] In this example it is easy to show how the series behaves at zero.

; then the partial sums of the Fourier series converge to the function with speed[10] for a constant

A function ƒ has an absolutely converging Fourier series if If this condition holds then

The family of all functions with absolutely converging Fourier series is a type of Banach algebra called the Wiener algebra, after Norbert Wiener, who proved that if ƒ has absolutely converging Fourier series and is never zero, then 1/ƒ has absolutely converging Fourier series.

A simplification of the original proof of Wiener's theorem was given by Israel Gelfand and later by Donald J. Newman in 1975.

Notice that the 1/2 here is essential—there is an example of a 1/2-Hölder functions due to Hardy and Littlewood,[14] which do not belong to the Wiener algebra.

Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only

If the partial sum SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ p < ∞.

The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s.

His result, now known as Carleson's theorem, tells the Fourier expansion of any function in L2 converges almost everywhere.

Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a continuous function ƒ such that the Fourier series of ƒ fails to converge on any point of E. Does the sequence 0,1,0,1,0,1,... (the partial sums of Grandi's series) converge to ⁠1/2⁠?

we denote the kth partial sum: It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it.

Fejér's theorem states that the above sequence of partial sums converge uniformly to ƒ.

The fact that for some constant c we have is quite clear when one examines the graph of Dirichlet's kernel.

For any continuous function f and any t one has However, for any order of growth ω(n) smaller than log, this no longer holds and it is possible to find a continuous function f such that for some t, The equivalent problem for divergence everywhere is open.

Sergei Konyagin managed to construct an integrable function such that for every t one has It is not known whether this example is best possible.

The only bound from the other direction known is log n. Upon examining the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses.

For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of

In particular, the equivalent of Carleson's theorem is still open for circular partial sums.