The generalized sampling theorem for transforms of not necessarily square integrable functions (Q1067153)

The generalized sampling theorem due to \textit{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).