On the degree of convergence of Laplace’s series

Abstract

t Lebesgue first showed the fundamental importance of the corresponding constants for Fourier's series; see Lésons sur les series trigonométriques, Paris, 1906, § 45. t Jackson shows (1.c., pp.306-309) that the corresponding maximum for Legendre's series in the interval ( 6) is inferior to a constant multiple of log n .§ The first proof of this relation may be found in § 2 of my paper Über die Laplace'sehe Reihe (Mathematische Annalen, vol.74 (1913), pp.213-270); the present proof contains several simplifications and also some new developments which will be used in § 4. * Dunham Jackson, II.Über die Genauigkeit der Annäherung stetiger