On the degree of convergence of the development of a continuous function according to Legendre’s polynomials

Abstract

It is the purpose of this paper to demonstrate three theorems and two corollaries, whoseformal statement will be preceded by a brief explanation. The treatment is suggested bv recent papers of LEsEsGuE,t in which corresponding problems are discussed for the case of trigonometric series. A part of the work of LEBESGUE is reproduced at length in my thesis,: and where the developments of the present paper are closely parallel to those in the trigonometric case I shall occasionally refer the reader to one of those other presentations.