Singular Integrals on the Line Group

Abstract

The material of this chapter is in many points a straightforward adaptation of the periodic counterparts of Chapter 1. This is particularly true for Sec. 3.1 which corresponds to Sec. 1.1, 1.3, thus treating convergence in the norms of the spaces X(ℝ). Special emphasis is placed upon the study of singular integrals of Fejer’s type. In Sec. 3.1.2 to each approximate identity on the real line a periodic approximate identity is associated via (3.1.28), (3.1.55), respectively. Important examples of singular integrals such as those of Fejer, Gauss-Weierstrass, and Cauchy-Poisson are introduced. Sec. 3.2 deals with pointwise convergence of convolution integrals, the results correspond to those of Sec. 1.4. Sec. 3.3 is concerned with nonperiodic counterparts of Sec. 1.5, thus with questions on the order of approximation by positive singular integrals on the real line. The method of test functions is touched upon and certain asymptotic expansions are given. Furthermore, Nikolskiĭ constants for periodic singular integrals of Fejer’s type with respect to Lipschitz classes are determined; these complete the results of Sec. 1.6.3 in the fractional case. Sec. 3.4 deals with direct approximation theorems for singular integrals, the kernels of which need not necessarily be positive. In case the order of approximation is O(ρ-∝), 0 < ∝⩽ 2, the results correspond to those of Sec. 1.6. For applications of these concerning higher order approximation we refer to Sec. 6.4 where certain periodic counterparts are also formulated. In Sec. 3.5 inverse approximation theorems for singular integrals of Fejer’s type are given. The proofs follow by a direct adaptation of Bernstein’s idea, already familiar from Sec. 2.3, 2.5. In Sec. 3.6 some aspects concerning shape preserving properties of approximation processes are discussed.