On the Representation of a Function by a Trigonometric Series

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Excerpt from On the Representation of a Function by a Trigonometric Series: Dissertation Presented to the Board of University Studies of the Johns Hopkins University for the Degree of Doctor of PhilosophyIn this thesis I have considered the representation, in trigonometric series, of a function of a real variable only.Since this subject has already been so fully treated I could hardly expect to obtain many new results. Accordingly it has been my special aim, after carefully examining the work of others, to investigate independently certain phases of the question, and thereby to obtain if possible a simpler development of the subject. To some extent my efforts have been rewarded, since, in several cases, I have arrived at results by methods considerably shorter than those which others have employed.The first two sections of this paper are introductory. Section 1 is purely historical, giving a brief account of the origin of the question and an outline of the principal work done upon it up to the present time.* In section 2 is pre sented the theorem of du bois-reymond which gives directly the form of the coefficients in the trigonometric series and proves for all cases the uniqueness of the development.In the next two sections I have shown that the convergence of this develop ment of an integrable function f (x) to f (5170) for w mo, depends only upon the behavior of the function in the vicinity of me. This is followed in sections 5 and 6 by a proof that the series thus converges to f (930) When f (cc), in the neighborhood of mg, is finite, possesses only a finite number of discontinuities and only a finite number of maxima and minima.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.