Author:
View More View Less
• 1 Eötvös Loránd University Department of Numerical Analysis Pázmány P. sétány I/C H-1117 Budapest Hungary
Restricted access

Cross Mark

This paper is devoted to the study of Θ-summability of Fourier-Jacobi series. We shall construct such processes (using summations) that are uniformly convergent in a Banach space (
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } } ,\parallel \cdot \parallel _{w_{\gamma ,\delta } }$$ \end{document}
) of continuous functions. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. Our aim is to give such conditions with respect to Jacobi weights wγ,δ , wα,β and to summation matrix Θ for which the uniform convergence holds for all f
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } }$$ \end{document}
. Order of convergence will also be investigated. The results and the methods are analogues to the discrete case (see [16] and [17]).
• Agahanov, S. A. and Natanson, G. I. , The Lebesgue function of Fourier-Jacobi sums, Vestnik Leningrad. Univ. , 23 (1968), no. 1, 11–23 (in russian). MR37 #478

Natanson G. I. , 'The Lebesgue function of Fourier-Jacobi sums ' (1968 ) 23 Vestnik Leningrad. Univ. : 11 -23.

• Chanillo, S. and Muckenhoupt, B. , Weak Type Estimates for Cesaro Sums of Jacobi Polynomial Series, Mem. Amer. Math. Soc. , 102 (1993), No. 487. MR93g :42018

• Felten, M. , Boundedness of first order Cesàro means in Jacobi spaces and weighted approximation on [−1, 1], 2004, Habilitationsschrift, Seminarberichte aus dem Fachbereich Mathematik der FernUniversität in Hagen (ISSN 0944-5838), Band 75, pp. 1–170.

• Felten, M. , Uniform boundedness of ( C; 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas), Acta Math. Hungar. , 118 (2008), no. 2, 227–263. MR2009c :42059

Felten M. , 'Uniform boundedness of (C; 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas) ' (2008 ) 118 Acta Math. Hungar. : 227 -263.

• Felten, M. , Uniform boundedness of ( C; 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations), Acta Math. Hungar. , 118 (2008). no. 3, 265–297.

Felten M. , 'Uniform boundedness of (C; 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations) ' (2008 ) 118 Acta Math. Hungar. : 265 -297.

• Lubinsky, D. S. and Totik, V. , Best weighted polynomial approximation via Jacobi expansions, SIAM J. Math. Anal. , 25 (1994), no. 2, 555–570. MR95d :41028

Totik V. , 'Best weighted polynomial approximation via Jacobi expansions ' (1994 ) 25 SIAM J. Math. Anal. : 555 -570.

• Natanson, I. P. , Constructive Function Theory , Akadémiai Kiadó (Budapest, 1952) (in Hungarian).

Natanson I. P. , '', in Constructive Function Theory , (1952 ) -.

• Osilenker, B. , Fourier Series in Orthogonal Polynomials , World Sci. Publ., Singapore — New Jersey — London — Hong Kong (1999). MR2001h :42001

Osilenker B. , '', in Fourier Series in Orthogonal Polynomials , (1999 ) -.

• Stechkin, S. B. , The approximation of periodic functions by Fejér sums (in Russian), Trudy Math. Inst. Steklov , 62 (1961), 48–60.

Stechkin S. B. , 'The approximation of periodic functions by Fejér sums (in Russian) ' (1961 ) 62 Trudy Math. Inst. Steklov : 48 -60.

• Suetin, P. K. , Classical Orthogonal Polynomials , Nauka, Moscow, 1979 (in Russian). MR80h :33001

Suetin P. K. , '', in Classical Orthogonal Polynomials , (1979 ) -.

• Szabados, J. , Weighted error estimates for approximation by Cesàro means of Fourier-Jacobi series in spaces of locally continuous functions, Anal. Math. , 34 (2008), no. 1, 59–69. MR2009c :40026

Szabados J. , 'Weighted error estimates for approximation by Cesàro means of Fourier-Jacobi series in spaces of locally continuous functions ' (2008 ) 34 Anal. Math. : 59 -69.

• Szegő, G. , Orthogonal Polynomials , AMS Coll. Publ., Vol. 23, Providence, 1975. MR51 #8724

• Szili, L. , On summability of weighted Lagrange interpolation on the roots of Jacobi polynomials, Acta Math. Hungar. , 99 (2003), no. 3, 209–231. MR2004a :41003

Szili L. , 'On summability of weighted Lagrange interpolation on the roots of Jacobi polynomials ' (2003 ) 99 Acta Math. Hungar. : 209 -231.

• Szili, L. , Cesàro summability of Lagrange interpolation on Jacobi roots, Studia Sci. Math. Hungar. , 41 (2004), no. 4, 437–451. MR2005m :40010

Szili L. , 'Cesàro summability of Lagrange interpolation on Jacobi roots ' (2004 ) 41 Studia Sci. Math. Hungar. : 437 -451.

• Szili, L. and Vértesi, P. , On uniform convergence of sequences of certain linear operators, Acta Math. Hungar. , 91 (2001), no. 1–2, 159–186. MR2004d :42010

Vértesi P. , 'On uniform convergence of sequences of certain linear operators ' (2001 ) 91 Acta Math. Hungar. : 159 -186.

• Szili, L. and Vértesi, P. , On summability of weighted Lagrange interpolation I. (General weights), Acta Math. Hungar. , 101 (2003), no. 4, 323–344. MR2004k :41003

Vértesi P. , 'On summability of weighted Lagrange interpolation I. (General weights) ' (2003 ) 101 Acta Math. Hungar. : 323 -344.

• Szili, L. and Vértesi, P. , On summability of weighted Lagrange interpolation III. (Jacobi weights), Acta Math. Hungar. , 104 (2004), no. 1–2, 39–62. MR2005i :41004

Vértesi P. , 'On summability of weighted Lagrange interpolation III. (Jacobi weights) ' (2004 ) 104 Acta Math. Hungar. : 39 -62.

All Time Past Year Past 30 Days
Abstract Views 9 9 0
Full Text Views 5 4 0