Thesis (doctoral)--Friedrich-Wilhelms Universität zu Berlin, 1922. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Johns Hopkins University, 1894. / Biographical sketch.

Part of Thesis--University of Michigan. / "Reprinted from Quarterly of applied mathematics, vol. V, no. 3, October, 1947.

Thesis (Ph. D.)--Johns Hopkins University, 1894. / Biographical sketch.

Let Wn be the n th partial sum of a Walsh series. There is a necessary and sufficient condition for pointwise convergence of the sequence W₂n , (n = 0, 1, 2,…), namely, [ Formulas omitted ] The aim of this thesis is to formulate and discuss a similar convergence equivalence for trigonometric series. / Science, Faculty of / Mathematics, Department of / Graduate

Call number: LD2668 .T4 1938 G71

Summary A result is discussed which permits the summing of series whose terms have more complicated sign patterns than simply alternating plus and minus. The Alternating Series Test, commonly taught in beginning calculus courses, is a corollary. This result, which is not difficult to prove, widens the series summable by beginning students and paves the way for understanding more advanced questions such as convergence of Fourier series. An elementary exposition is given of Dirichlet’s Test for the convergence of a series and an elementary example suitable for a beginning calculus class and a more advanced example involving a Fourier series which is appropriate for an advanced calculus class are provided. Finally, two examples are discussed for which Dirichlet’s Test does not apply and a general procedure is given for deciding the convergence or divergence of these and similar examples.

Summary Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these nonelementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.

The purpose of this paper is to develop some of the more basic Fourier transforms which are the outgrowth of the Fourier theorem. Although often approached from the stand-point of the series, this paper will approach the theorem from the standpoint of the integral.

This thesis is intended as an introduction to the study of one type of trigonometric series, the Fourier series.