Summability kernel explained

In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary,^[1] but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.

Definition

Let

T : = R / Z . A summability kernel is a sequence (k _{n)} in L ^{1(T)} that satisfies\int _{Tk} _{n(t)dt=1} \int _{T|k} _{n(t)|dt\le} M (uniformly bounded) \int _{\delta\le|t|\le (1)/(2)} |k _{n(t)|dt\to0} as n\to inf ty , for every \delta> 0 .

Note that if

k _{n\ge0} for all n , i.e. (k _{n)} is a positive summability kernel, then the second requirement follows automatically from the first.

With the more usual convention

T = R / 2 \pi Z , the first equation becomes (1)/(2 \pi) \int _{Tk} _{n(t)dt=1} , and the upper limit of integration on the third equation should be extended to \pi , so that the condition 3 above should be \int _{\delta\le|t|\le\pi} |k _{n(t)|dt\to0} as n\to inf ty , for every \delta> 0 .

This expresses the fact that the mass concentrates around the origin as

n increases.

One can also consider

R rather than T ; then (1) and (2) are integrated over R , and (3) over |t|>\delta .

Examples

• The Fejér kernel
• The Poisson kernel (continuous index)
• The Landau kernel
• The Dirichlet kernel is not a summability kernel, since it fails the second requirement.

Convolutions

Let

(k _{n)} be a summability kernel, and * denote the convolution operation.

• If

(k _{n),f\inl{C}(T)} (continuous functions on T ), then k _{n*f\to} f in l{C}( T ) , i.e. uniformly, as n\to inf ty . In the case of the Fejer kernel this is known as Fejér's theorem.

• If

(k _{n),f\in} L ^{1(T)} , then k _{n*f\to} f in L ^{1(T)} , as n\to inf ty .

• If

(k _{n)} is radially decreasing symmetric and f\in L ^{1(T)} , then k _{n*f\to} f pointwise a.e., as n\to inf ty . This uses the Hardy–Littlewood maximal function. If (k _{n)} is not radially decreasing symmetric, but the decreasing symmetrization \widetilde{k} _{n(x):=\sup} _{|y|\ge|x|} k _{n(y)} satisfies \sup _{n\inN} \|\widetilde{k} _{n\|} _{1<infty} , then a.e. convergence still holds, using a similar argument.

Notes and References

1. Book: Pereyra . María . Ward . Lesley . Harmonic Analysis: From Fourier to Wavelets . 2012 . American Mathematical Society . 90 .