## Doctoral Dissertations

#### Date of Award

12-1958

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

F. A. Ficken

#### Committee Members

Leo Simons, Walter Snyder, O. M. Harrod, D. D. Lillian

#### Abstract

Introduction: Many of the properties of the ordinary Fourier series expansion of a given function are shared by the orthogonal expansion in terms of eigenfunctions of a second order ordinary differential operator. Let p = p(x) and q = q(x) be real-valued functions such that p, p', and q are continuous, and p(x) > 0, on a finite interval a ≤ x ≤ b. Let λ be a complex parameter. The classical Strum-Liouville theory [9, section 27; 4, Chapter 7; 21, Chapter 1]^{1} is concerned with solutions of the differential equation -(py') + qy = λ, which satisfy certain real boundary conditions whose form need not be given here. These solutions, the so-called eigenfunctions, exist only for certain values of λ, the corresponding eigenvalues constitute a countable set of real numbers which cluster only at ∞. The corresponding eigenfunctions constitute an orthogonal system on [a,b] which is complete in L^{2}(a, b). Thus the Parseval relation is also valid.

#### Recommended Citation

Oehring, Charles C., "Spectral Theory of Self-Adjoint Ordinary Differential Operators. " PhD diss., University of Tennessee, 1958.

https://trace.tennessee.edu/utk_graddiss/2951