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Zapiski Nauchnykh Seminarov POMI, 2011, Volume 389, Pages 34–57 (Mi znsl4117)  

This article is cited in 5 scientific papers (total in 5 papers)

On the norms of generalized translation operators generated by Jacobi–Dunkl operators

O. L. Vinogradov

Saint-Petersburg State University, Saint-Petersburg, Russia
Full-text PDF (668 kB) Citations (5)
References:
Abstract: We establish an integral representation and improve the norm estimate for the generalized translation operators generated by Jacobi–Dunkl operators
$$ \Lambda_{\alpha,\beta}f(x)=f'(x)+\frac{A_{\alpha,\beta}'(x)}{A_{\alpha,\beta}(x)}\,\frac{f(x)-f(-x)}2, $$
where
$$ A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|\sin x|, $$
in the spaces $L_p[-\pi,\pi]$ with the weight $A_{\alpha,\beta}$. For $\alpha\ge\beta\ge-\frac12$ we prove that these norms do not exceed $2$.
Key words and phrases: Jacobi polynomials, generalized translation operator, Jacobi–Dunkl operator.
Received: 11.05.2011
English version:
Journal of Mathematical Sciences (New York), 2012, Volume 182, Issue 5, Pages 603–616
DOI: https://doi.org/10.1007/s10958-012-0765-8
Bibliographic databases:
Document Type: Article
UDC: 517.5
Language: Russian
Citation: O. L. Vinogradov, “On the norms of generalized translation operators generated by Jacobi–Dunkl operators”, Investigations on linear operators and function theory. Part 39, Zap. Nauchn. Sem. POMI, 389, POMI, St. Petersburg, 2011, 34–57; J. Math. Sci. (N. Y.), 182:5 (2012), 603–616
Citation in format AMSBIB
\Bibitem{Vin11}
\by O.~L.~Vinogradov
\paper On the norms of generalized translation operators generated by Jacobi--Dunkl operators
\inbook Investigations on linear operators and function theory. Part~39
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 389
\pages 34--57
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl4117}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 182
\issue 5
\pages 603--616
\crossref{https://doi.org/10.1007/s10958-012-0765-8}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84860358744}
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  • https://www.mathnet.ru/eng/znsl/v389/p34
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    References:34
     
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