Problems in the theory of approximation of functions on an arbitrary compact rank-one symmetric space  in the metric of , , are investigated. The approximating functions are generalized spherical polynomials, that is, linear combinations of eigenfunctions of the Beltrami-Laplace operator on . Analogues of the direct Jackson theorems are proved for the modulus of smoothness (of arbitrary order) constructed by using the operator of spherical averaging. It is established that the modulus of smoothness and the -functional constructed from the Sobolev-type space corresponding to the Beltrami-Laplace differential operator are equivalent. Bibliography: 35 titles.

Some problems in the theory of approximation of functions on compact homogeneous manifolds

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О некоторых задачах теории приближения функций на компактных однородных многообразиях@@@Some problems in the theory of approximation of functions on compact homogeneous manifolds

Classes of functions in the space of continuous functions defined on the interval and vanishing at its end-points are described for which there is pointwise and approximate uniform convergence of Lagrange-type operators These operators involve the solutions of the Cauchy problem for the equation where (here is the ball of radius in the space of functions of bounded variation vanishing at the origin, and ). Several modifications of this operator are proposed, which allow an arbitrary continuous function on to be approximated uniformly. Bibliography: 40 titles.

A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval

Спектральный синтез на группе конформных автоморфизмов единичного круга@@@Spectral synthesis on the group of conformal automorphisms of the unit disc

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Обобщение теоремы отсчетов Уиттекера - Котельникова - Шеннона для непрерывных функций на отрезке@@@A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval

Invariant subspaces in certain function spaces on the simplest solvable lie group

Using the Fourier–Walsh transform on ℝ+ = [0,+∞), we prove a dyadic analog of the classical Titchmarsh theorem on the description of the image under the Fourier transformation of the set of functions satisfying the Lipschitz condition in L2.

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

Let 
$$f(x)$$
 be a function belonging to the Lebesgue class 
$$L^p({\mathbb R}_+)$$
 on the semiaxis 
$${\mathbb R}_+=[0,+\infty)$$
, 
$$1\le p\le 2$$
, and let 
$$\widehat{f}$$
 be the Fourier–Walsh transform of the function 
$$f$$
. In this paper, we give the solution of the following problem: if the function 
$$f$$
 belongs to the dyadic Dini–Lipschitz class 
$$\operatorname{DLip}_\oplus(\alpha,\beta,p;{\mathbb R}_+)$$
, 
$$\alpha>0$$
, 
$$\beta\in{\mathbb R}$$
, then for what values of 
$$r$$
 can we guarantee that 
$$\widehat{f}$$
 belongs to 
$$L^r({\mathbb R}_+)$$
? The result obtained is an analog of the classical Titchmarsh theorem on the Fourier transform of functions from Lipschitz classes on 
$${\mathbb R}$$
.

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Let M be an arbitrary compact Riemannian symmetric space of rank 1. The function spaces 
$$H_p^r$$

 of Nikol'skii type were introduced earlier by means of averaged differences along geodesics. In the present paper we give an equivalent description of these spaces and norms in them by using the Laplace--Beltrami operator. The results obtained generalize the results of Nikol'skii and Lizorkin on the spaces 
$$H_p^r$$

 over the sphere.

On Equivalent Normings in the Spaces H p M on Compact Homogeneous Spaces

In this paper, we obtain an analog of the Whittaker-Kotel’nikov-Shannon theorem on the basis of the harmonic Fourier-Bessel analysis. We construct a new interpolation formula for a class of entire functions of exponential type in which generalized Bessel translations are used instead of the usual translations.

S. S. Platonov

Papers

Invariant subspaces in certain function spaces on the simplest solvable lie group

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

On Equivalent Normings in the Spaces H p M on Compact Homogeneous Spaces

Analog of the Whittaker-Kotel’nikov-Shannon theorem from the point of view of Fourier-Bessel analysis