# Theorems of Wiener-Lévy type for integral operators in C_{p}

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## Abstract

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_{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.