Geodesic
On the norms of generalized translation operators generated by Jacobi–Dunkl operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 34-57 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $2$. 
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     author = {O. L. Vinogradov},
     title = {On the norms of generalized translation operators generated by {Jacobi{\textendash}Dunkl} operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--57},
     year = {2011},
     volume = {389},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/}
}
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O. L. Vinogradov. On the norms of generalized translation operators generated by Jacobi–Dunkl operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 34-57. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/

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