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The generalized sampling theorem for transforms of not necessarily square integrable functions. (English) Zbl 0579.44002
The generalized sampling theorem due to H. P. Kramer [J. Math. Phys. 38, 68-72 (1959; Zbl 0196.317)] is the following: ”Let I be an interval. Suppose that for each $$t\in {\mathbb{R}}$$, $$f(t)=\int_{I}K(t,x) g(x) dx$$, where $$g(x)\in L^ 2(I)$$, $$K(t,x)\in L^ 2(I)$$ and $$\{K(t_ n,x)\}$$ is a complete orthogonal set on $$L^ 2(I)$$. Then $$f(t)=\lim_{N\to \infty}\sum_{| n| \leq N}f(t_ n) S_ n(t)$$, where $$S_ n(t)$$ is the Fourier coefficient of the kernel K(t,x) in terms of the complete orthogonal set $$\{K(t_ n,x)\}.''$$
This note presents the following extension: The above generalized sampling expansion is valid for $$g(x)\in L^ p(a,b)$$, where $$1\leq p\leq 2$$, for all differentiable kernels K(,x).
Reviewer: M.Voicu

##### MSC:
 44A15 Special integral transforms (Legendre, Hilbert, etc.) 93C57 Sampled-data control/observation systems
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