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.
Definition Let T := R / Z {\displaystyle \mathbb {T} :=\mathbb {R} /\mathbb {Z} } . A summability kernel is a sequence ( k n ) {\displaystyle (k_{n})} in L 1 ( T ) {\displaystyle L^{1}(\mathbb {T} )} that satisfies
∫ T k n ( t ) d t = 1 {\displaystyle \int _{\mathbb {T} }k_{n}(t)\,dt=1}
∫ T | k n ( t ) | d t ≤ M {\displaystyle \int _{\mathbb {T} }|k_{n}(t)|\,dt\leq M} (uniformly bounded)
∫ δ ≤ | t | ≤ 1 2 | k n ( t ) | d t → 0 {\displaystyle \int _{\delta \leq |t|\leq {\frac {1}{2}}}|k_{n}(t)|\,dt\to 0} as n → ∞ {\displaystyle n\to \infty } , for every δ > 0 {\displaystyle \delta >0} . Note that if k n ≥ 0 {\displaystyle k_{n}\geq 0} for all n {\displaystyle n} , i.e. ( k n ) {\displaystyle (k_{n})} is a positive summability kernel, then the second requirement follows automatically from the first. With the more usual convention T = R / 2 π Z {\displaystyle \mathbb {T} =\mathbb {R} /2\pi \mathbb {Z} } , the first equation becomes 1 2 π ∫ T k n ( t ) d t = 1 {\displaystyle {\frac {1}{2\pi }}\int _{\mathbb {T} }k_{n}(t)\,dt=1} , and the upper limit of integration on the third equation should be extended to π {\displaystyle \pi } , so that the condition 3 above should be
∫ δ ≤ | t | ≤ π | k n ( t ) | d t → 0 {\displaystyle \int _{\delta \leq |t|\leq \pi }|k_{n}(t)|\,dt\to 0} as n → ∞ {\displaystyle n\to \infty } , for every δ > 0 {\displaystyle \delta >0} . This expresses the fact that the mass concentrates around the origin as n {\displaystyle n} increases. One can also consider R {\displaystyle \mathbb {R} } rather than T {\displaystyle \mathbb {T} } ; then (1) and (2) are integrated over R {\displaystyle \mathbb {R} } , and (3) over | t | > δ {\displaystyle |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 ) {\displaystyle (k_{n})} be a summability kernel, and ∗ {\displaystyle *} denote the convolution operation.
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