On lower bounds for generalized Lebesgue constants of Fourier-Jacobi sums in $$$L^p_{\rho}$$$ spaces

S.V. Goncharov

doi:10.15421/241011

On lower bounds for generalized Lebesgue constants of Fourier-Jacobi sums in $$$L^p_{\rho}$$$ spaces

S.V. Goncharov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0002-8071-8746

Abstract

We obtain lower estimates for the generalized Lebesgue constants of Fourier-Jacobi sums, in the spaces of functions being integrable with the $$$\rho(t) = (1-t)^A (1+t)^B$$$ weight, therefore confirming an exactness of known upper estimates.

Keywords

Jacobi polynomials; Fourier-Jacobi sums; Lebesgue constants

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References

Muckenhoupt B. "Mean convergence of Hermite and Laguerre series. II", Trans. Amer. Math. Soc., 1970; 147(2): pp. 433-460. doi:10.2307/1995205

Muckenhoupt B. "Mean convergence of Jacobi series", Proc. Amer. Math. Soc., 1969; 23(2): pp. 306-310. doi:10.2307/2037162

Newman J., Rudin W. "Mean convergence of orthogonal series", Proc. Amer. Math. Soc., 1952; 3(2): pp. 219-222. doi:10.2307/2032262

Pollard H. "The mean convergence of orthogonal series. III", Duke Math. J., 1949; 16(1): pp. 189-191. doi:10.1215/S0012-7094-49-01619-1

Szego G. Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. Vol. XXIII, 1959.

Badkov V.M. "Approximation properties of Fourier series in orthogonal polynomials", Uspekhi matem. nauk, 1978; 33(4): pp. 51-106.

Kazakova N.M. "On orders of Lebesgue constants of Fourier-Jacobi sums in $$$L^r_{\rho}$$$ spaces", Depon. in VINITI, 1981.

Motornyi V.P. "On mean convergence of Fourier series by Legendre polynomials", Izv. AN sssr. Ser. Matem., 1973; 37: pp. 135-147.

Motornyi V.P., Goncharov S.V., Nitiema P.K. "On mean convergence of Fourier-Jacobi series", Dopovidi NAN Ukrainy, 2010; 3: pp. 35-39.