Differential and falsified sampling expansions

Abstract

Differential and falsified sampling expansions ${\sum }_{k\in {\mathbb{Z}}^d}{c}_k\varphi (M^{j}x+k)$, where $M$ is a matrix dilation, are studied. In the case of differential expansions, $c_k=Lf(M^{-j}\cdot )(-k)$, where $L$ is an appropriate differential operator. For a large class of functions $\varphi$, the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of $\varphi$ from this class is required. In the present paper, we obtain similar results for a class of band-limited functions $\varphi$ with the discontinuous Fourier transform. In the case of falsified expansions, $c_k$ is the mathematical expectation of random integral average of a signal $f$ near the point $M^{-j}k$. To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in $L_p$-norm are given in terms of the Fourier transform of $f$.

arXiv admin note: substantial text overlap with arXiv:1407.0321

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