On the Representation of a Function by a Trigonometric Series: Dissertation Presented to the Board of University Studies of the Johns Hopkins ... of Doctor of Philosophy (Classic Reprint) - Softcover

Manning, Edward P.

Softcover

ISBN 10: 1333480881 ISBN 13: 9781333480882

Publisher: Forgotten Books, 2018

Synopsis

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 Philosophy

In 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.

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Edition details

• Publisher: Forgotten Books
• Publication date: 2018
• Language: English
• ISBN 10: 1333480881
• ISBN 13: 9781333480882
• Binding: Paperback
• Number of pages: 37