Theorems of Wiener-Lévy type for integral operators in C_p

The classical theorem of N. Wiener and P. Lévy states that if f(x) has an absolutely convergent Fourier series and W(z) is an analytic function whose domain contains the range of f(x), then W[f(x)] also has an absolutely convergent Fourier series. The main result of this paper is an analog of the Wiener-Lévy Theorem in which we consider analytic transformations acting upon kernels of integral operators of the form (Tφ)(x) = ∫_-π^π K(x,y)φ(y)dy and the so-called s-numbers of K and W[K] take the place of the classical Fourier coefficients of f and W[f]. Theorem: Let W(z) be analytic in a region R and let K(x,y) map the square [-π,π] x [-π,π] continuously into R. If {φ_n}_n=1^∞ and {ψ_n}_n=1^∞ are a full set of continuous singular functions for K(x,y) and ∑_n=1^∞ [s_n (K)]^P | | φ_n | |_∞^P| | ψ<sub>n</sub | |_∞^P < ∞ for some 0 < p ≤ 1, then ∑_n=1^∞ [s_n (W[K])]^P < ∞ or, expressed alternatively, W[K] belongs to the Schatten class C_p. The classical Wiener-Lévy Theorem is obtained as a corollary in the special situation when p = 1 and K(x,y) ≡ f(x-y) is a difference kernel. For the case 1 < p < 2 we generalize a Fourier series result of L. Alpar to the following theorem in integral operator theory. Theorem: Let W(z) be analytic (but not necessarily single-valued) in a region R and let K(x,y) satisfy the following conditions: a) K(x,y) maps the square [-π,π] x [-π,π] continuously into R. b) K(x,y) is 2π-periodic in x (or y). c) K(x,y) satisfies an integrated Lipschitz condition of order α relatively uniformly in x (or y) where p⁻¹ < α ≤ 1, 1 < p < 2. Then, if W(z) returns to its initial determination after z travels completely around any curve C_y (or C_x) of the form z = K(x,y), -π ≤ x (or y) ≤ π, then W[K(x,y)] ε C_p. To round out the paper, we show that analytic transformations preserve smoothness conditions of Lipschitz and bounded variation type, and consequently we are able to give a number of sufficiency conditions for analytic functions of kernels to be in various kernel classes. Finally, we investigate the converse of the Wiener-Lévy Theorem and how analogues of it relate to integral operators. The paper concludes with suggestions of several interesting questions warranting further study.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1976

Author dc:creator

• Steel, Christopher Alan

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10919/77759

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/77759