A Note on Reconstruction of Bandlimited Signals of Several Variables Sampled at Nyquist Rate

Volume 30, Issue 3 (2019), pp. 529–552

Saulius Norvidas

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Pub. online: 1 January 2019
Type: Research Article
Open Access

Received

1 February 2019

1 February 2019

Accepted

1 August 2019

1 August 2019

Published

1 January 2019

1 January 2019

#### Abstract

A standard problem in certain applications requires one to find a reconstruction of an analogue signal

*f*from a sequence of its samples $f{({t_{k}})_{k}}$. The great success of such a reconstruction consists, under additional assumptions, in the fact that an analogue signal*f*of a real variable $t\in \mathbb{R}$ can be represented equivalently by a sequence of complex numbers $f{({t_{k}})_{k}}$, i.e. by a digital signal. In the sequel, this digital signal can be processed and filtered very efficiently, for example, on digital computers. The sampling theory is one of the theoretical foundations of the conversion from analog to digital signals. There is a long list of impressive research results in this area starting with the classical work of Shannon. Note that the well known Shannon sampling theory is mainly for one variable signals. In this paper, we concern with bandlimited signals of several variables, whose restriction to Euclidean space ${\mathbb{R}^{n}}$ has finite*p*-energy. We present sampling series, where signals are sampled at Nyquist rate. These series involve digital samples of signals and also samples of their partial derivatives. It is important that our reconstruction is stable in the sense that sampling series converge absolutely and uniformly on the whole ${\mathbb{R}^{n}}$. Therefore, having a stable reconstruction process, it is possible to bound the approximation error, which is made by using only of the partial sum with finitely many samples.#### References

Butzer, Ferreira, P.L., S. G, P.J., Higgins, J.R., Schmeisser, G., Stens, R.L. (2011). The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections. Applicable Analysis, 90(3–4), 431–461.

Lin, R. (2019). An optimal convergence rate for the Gaussian regularized shannon sampling series.

*Numerical Functional Analysis and Optimization*, 1–19. https://doi.org/10.1080/01630563.2018.1549072.#### Biographies

Norvidas Saulius

**S. Norvidas** is a principal researcher of Vilnius university Institute of Data Science and Digital Technologies. His fields of interest are harmonics analysis, functions theory of real and complex variables, functional analysis and operator theory.