Integrability Theorems for Trigonometric Transforms

The Hausdorff Mean of a Fourier-Stieltjes Transform

“…As a consequence, we obtain: (a) certain improved variants of the formulas of Fejer and Wiener for the inversion and quadratic variation of Fourier-Stieltjes transforms (Corollary 1); (b) a strengthened generalization of the mean ergodic theorem for a one-parameter group of unitary operators. Theorem 2 establishes the uniqueness of the generalized Fourier transforms introduced by Sz.-Nagy [14] and studied extensively in [1].…”

“…When the kernel ip is the characteristic function of [0,1 ], Tp reduces to the integral arithmetic average of p over [0, x]. The summability properties of T are well known [8, pp.…”

Integrability of trigonometric series. II

“…Let 0 < r < 1 and b n \0. Then the function g defined by g(t) -Σ*=i b n sin nt, satisfies the relation (1). The same holds for even functions.…”

“…( [1]), [4]) Let 0 < r < 1, then there is an odd function Sunouchi [6], Edmonds [3] and Boas [2] (cf. [1]) proved that THEOREM 3. Let 0 < r < 1 and b n \0.…”

Differentiable positive definite kernels and Lipschitz continuity

“…Surprisingly, it appears that this work with positive definite kernels represents another instance in which the kernel results actually predate the Fourier series results. Comparable conclusions do exist for Fourier sine and/or cosine series, notably Lorentz's theorem (see [7], p. 148; see also [1], §7). But Oehring's investigations [9,10] are, as far as we know, the first consideration of these questions for classical exponential Fourier series.…”