On the Best Approximation for Singular Integrals by Laplace — Transform Methods

Abstract

It is the aim of this paper to continue the investigations on the degree of the best approximation of functions by singular integrals with positive kernels. One of the authors first treated these problems with semi-group theory [2, 3], and later introduced integral transform methods²⁾, in particular Fourier-transform methods [4, 5, 6], in case the singular integrals are convolution integrals connected with the Fourier-transform. By the latter method one can generalize the results in two respects: The function space under consideration need not be reflexive, a condition very useful in the semi-group approach, and the singular integrals need not be semi-group operators. In this paper Laplace-transform methods are used to treat problems of the type where the singular integrals are classical convolution integrals connected with the Laplace-transform. We shall also see that the latter method has the same advantages mentioned above over the semi-group approach as has the Fourier-transform method. Although there are connections between the Fourier-and Laplace-transform methods, it may be mentioned that the special properties and peculiar structure of the Laplace-transform play an important role in the proofs and the formulations of the stated theorems³⁾.

The lecture was presented by H. Berens.