Asymptotics of approximation of functions by conjugate Poisson integrals

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.

A symmetrized conjugacy scheme for orthogonal expansions

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian $\mathbb{L}$ to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing $\mathbb{L}$ are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

On Hankel conjugate functions

We consider some aspects of harmonic analysis of the differential operator . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform Hν. We present a fairly detailed analysis of the corresponding Poisson semigroup {Pt}t > 0: this is given in a weighted setting with Ap-weights involved. Then, we consider conjugate Poisson integrals of functions from Lp(w), w ∈ Ap, 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in Lp(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform Hν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.