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 Mat. Zametki, 2013, Volume 94, Issue 2, Pages 295–309 (Mi mz10292)

Limit Ultraspherical Series and Their Approximation Properties

I. I. Sharapudinov

Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala

Abstract: We study new series of the form $\sum_{k=0}^\infty f_k^{-1} \widehat P_k^{-1}(x)$ in which the general term $f_k^{-1}\widehat P_k^{-1}(x)$, $k=0,1,…$, is obtained by passing to the limit as $\alpha\to-1$ from the general term $\widehat f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ of the Fourier series $\sum_{k=0}^\infty f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ in Jacobi ultraspherical polynomials $\widehat P_k^{\alpha,\alpha}(x)$ generating, for $\alpha>-1$, an orthonormal system with weight $(1-x^2)^\alpha$ on $[-1,1]$. We study the properties of the partial sums $S_n^{-1}(f,x)=\sum_{k=0}^nf_k^{-1}\widehat P_k^{-1}(x)$ of the limit ultraspherical series $\sum_{k=0}^\infty f_k^{-1}\widehat P_k^{-1}(x)$. In particular, it is shown that the operator $S_n^{-1}(f)=S_n^{-1}(f,x)$ is the projection onto the subspace of algebraic polynomials $p_n=p_n(x)$ of degree at most $n$, i.e., $S_n(p_n)=p_n$; in addition, $S_n^{-1}(f,x)$ coincides with $f(x)$ at the endpoints $\pm1$, i.e., $S_n^{-1}(f,\pm1)=f(\pm1)$. It is proved that the Lebesgue function $\Lambda_n(x)$ of the partial sums $S_n^{-1}(f,x)$ is of the order of growth equal to $O(\ln n)$, and, more precisely, it is proved that $\Lambda_n(x)\le c(1+\ln(1+n\sqrt{1-x^2}\mspace{2mu}))$, $-1\le x\le 1$.

Keywords: limit ultraspherical series, Fourier series, Lebesgue function of partial sums, Jacobi polynomial, Christoffel–Darboux formula, approximation of continuous functions.

DOI: https://doi.org/10.4213/mzm10292

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English version:
Mathematical Notes, 2013, 94:2, 281–293

Bibliographic databases:

UDC: 517.587

Citation: I. I. Sharapudinov, “Limit Ultraspherical Series and Their Approximation Properties”, Mat. Zametki, 94:2 (2013), 295–309; Math. Notes, 94:2 (2013), 281–293

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10292
• https://doi.org/10.4213/mzm10292
• http://mi.mathnet.ru/eng/mz/v94/i2/p295

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

This publication is cited in the following articles:
1. I. I. Sharapudinov, “Some special series in ultraspherical polynomials and their approximation properties”, Izv. Math., 78:5 (2014), 1036–1059
2. I. I. Sharapudinov, “Nekotorye spetsialnye dvumernye ryady po sisteme $\{\sin x\sin kx\}$ i ikh approksimativnye svoistva”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(1) (2014), 407–412
3. I. I. Sharapudinov, G. G. Akniev, “Diskretnye preobrazovaniya so svoistvom prilipaniya na osnove sistemy $\{\sin x\sin kx\}$ i sistemy polinomov Chebysheva vtorogo roda”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(1) (2014), 413–422
4. I. I. Sharapudinov, “Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$”, Sb. Math., 206:4 (2015), 600–617
5. M. S. Sultanakhmedov, “Spetsialnye veivlety na osnove polinomov Chebysheva vtorogo roda”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:1 (2016), 34–41
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