Some problems on expansion of functions in double Fourier-Bessel series. (Russian, English) Zbl 1114.41022

Zh. Vychisl. Mat. Mat. Fiz. 44, No. 12, 2128-2149 (2004); translation in Comput. Math. Math. Phys. 44, No. 12, 2024-2044 (2004).
Some questions concerning expansions of functions of two variables in double Fourier-Bessel series are considered. Sharp or order estimates are obtained for the rate of their convergence in the classes of functions characterized by generalized moduli of continuity of various orders introduced by the first author et al. [Comput. Math. Math. Phys. 44, No. 12, 2024–2044 (2004); translation from Zh. Vychisl. Mat. Mat. Fiz. 44, No. 12, 2128–2149 (2004; Zbl 1114.42004), Comput. Math. Math. Phys. 41, No. 11, 1557–1577 (2001); translation from Zh. Vychisl. Mat. Mat. Fiz. 41, No. 11, 1637–1657 (2001; Zbl 1031.42027)]. A relation between the rate of convergence and the smoothness of the function expanded is established. Sharp or weak equivalent estimates of Kolmogorov widths are obtained for the classes of functions under consideration. Sufficient conditions are presented for the absolute conver- gence of a Fourier-Bessel series, which plays an important role in validating the method of separation of variables in mathematical physics.


41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A45 Theoretical approximation of solutions to ordinary differential equations
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series