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, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.
Let T := R / Z . A summability kernel is a sequence ( k n ) in L 1 ( T ) that satisfies
- ∫ T k n ( t ) d t = 1
- ∫ T | k n ( t ) | d t ≤ M (uniformly bounded)
- ∫ δ ≤ | t | ≤ 1 2 | k n ( t ) | d t → 0 as n → ∞ , for every δ > 0 .
Note that if k n ≥ 0 for all n , i.e. ( k n ) is a positive summability kernel, then the second requirement follows automatically from the first.
If instead we take the convention T = R / 2 π Z , the first equation becomes 1 2 π ∫ T k n ( t ) d t = 1 , and the upper limit of integration on the third equation should be extended to π .
We can also consider R rather than T ; then we integrate (1) and (2) over R , and (3) over | t | > δ .
The Fejér kernelThe Poisson kernel (continuous index)The Dirichlet kernel is not a summability kernel, since it fails the second requirement.Let ( k n ) be a summability kernel, and ∗ denote the convolution operation.
If ( k n ) , f ∈ C ( T ) (continuous functions on T ), then k n ∗ f → f in C ( T ) , i.e. uniformly, as n → ∞ .If ( k n ) , f ∈ L 1 ( T ) , then k n ∗ f → f in L 1 ( T ) , as n → ∞ .If ( k n ) is radially decreasing symmetric and f ∈ L 1 ( T ) , then k n ∗ f → f pointwise a.e., as n → ∞ . This uses the Hardy–Littlewood maximal function. If ( k n ) is not radially decreasing symmetric, but the decreasing symmetrization k ~ n ( x ) := sup | y | ≥ | x | k n ( y ) satisfies sup n ∈ N ∥ k n ~ ∥ 1 < ∞ , then a.e. convergence still holds, using a similar argument.