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arXiv · 1703.10420

Differential and falsified sampling expansions

Yu. Kolomoitsev·A. Krivoshein·M. Skopina
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Abstract

Differential and falsified sampling expansions $\sum_{k\in \mathbb{Z}^d}c_k\phi(M^jx+k)$, 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 $\phi$, the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of $\phi$ from this class is required. In the present paper, we obtain similar results for a class of band-limited functions $\phi$ 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 $f$.

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Yu. Kolomoitsev, A. Krivoshein, M. Skopina. 2017-03-30. Differential and falsified sampling expansions. https://arxiv.org/abs/1703.10420

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