E. V. Mishchenko, “Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials”, Sibirsk. Mat. Zh., 51:4 (2010), 829–837; Siberian Math. J., 51:4 (2010), 660

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Sibirskii Matematicheskii Zhurnal, 2010, Volume 51, Number 4, Pages 829–837 (Mi smj2128)

This article is cited in 11 scientific papers (total in 11 papers)

Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials

E. V. Mishchenko^ab

^a Sobolev Institute of Mathematics, Novosibirsk, Russia
^b Novosibirsk State University, Novosibirsk, Russia

Full-text PDF (298 kB) Citations (11)

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Abstract: The problem of determining the upper and lower Riesz bounds for the mth order

B

$B$ -spline basis is reduced to analyzing the series

\sum_{j = - \infty}^{\infty} \frac{1}{(x - j)^{2 m}}

$\sum_{j=-\infty}^\infty\frac1{(x-j)^{2m}}$ . The sum of the series is shown to be a ratio of trigonometric polynomials of a particular shape. Some properties of these polynomials that help to determine the Riesz bounds are established. The results are applied in the theory of series to find the sums of some power series.

Keywords:

B

$B$ -spline, Riesz basis, upper and lower Riesz bounds, trigonometric polynomial, power series.

Received: 21.01.2009

English version:
Siberian Mathematical Journal, 2010, Volume 51, Issue 4, Pages 660–666
DOI: https://doi.org/10.1007/s11202-010-0067-7

Bibliographic databases:

Document Type: Article

UDC: 517.518.34+517.537.3

Language: Russian

Citation: E. V. Mishchenko, “Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials”, Sibirsk. Mat. Zh., 51:4 (2010), 829–837; Siberian Math. J., 51:4 (2010), 660–666

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/smj2128

https://www.mathnet.ru/eng/smj/v51/i4/p829

This publication is cited in the following 11 articles:

Kumari Priyanka, A. Antony Selvan, “Derivative Sampling Expansions in Shift-Invariant Spaces With Error Estimates Covering Discontinuous Signals”, IEEE Trans. Inform. Theory, 70:8 (2024), 5453
E. V. Mishchenko, “Stability condition and Riesz bounds for exponential splines”, Sib. elektron. matem. izv., 20:2 (2023), 1430–1442
A. Antony Selvan, Kumari Priyanka, “Perturbation Theorems for Regular Sampling in Wavelet Subspaces”, Acta Appl Math, 182:1 (2022)
Yu. S. Volkov, “Efficient computation of Favard constants and their connection to Euler polynomials and numbers”, Sib. elektron. matem. izv., 17 (2020), 1921–1942
Kumar A., Sampath S., “Average Sampling and Reconstruction in Shift-Invariant Spaces and Variable Bandwidth Spaces”, Appl. Anal., 99:4 (2020), 672–699
De Carli L., Vellucci P., “P-Riesz Bases in Quasi Shift Invariant Spaces”, Frames and Harmonic Analysis, Contemporary Mathematics, 706, eds. Kim Y., Narayan S., Picioroaga G., Weber E., Amer Mathematical Soc, 2018, 201–213
De Carli L., Vellucci P., “Stability Results For Gabor Frames and the P-Order Hold Models”, Linear Alg. Appl., 536 (2018), 186–200
Selvan A.A., Radha R., “Sampling and Reconstruction in Shift Invariant Spaces of -Spline Functions”, Acta Appl. Math., 145:1 (2016), 175–192
V. G. Alekseev, V. A. Sukhodoev, “Schoenberg’s polynomial B-splines of odd degrees: A brief review of application”, Comput. Math. Math. Phys., 52:10 (2012), 1331–1341
Alekseev V.G., Sukhodoev V.A., “Polinomialnye b-splainy shenberga i ikh primenenie k postroeniyu ortogonalnykh veivlet-sistem”, Elektromagnitnye volny i elektronnye sistemy, 17:4 (2012), 17–20
V. G. Alekseev, V. A. Sukhodoev, “Schoenberg's polynomial B-splines of odd degrees: A brief review of application”, Comput. Math. and Math. Phys., 52:10 (2012), 1331

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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