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Abstract: We investigate symmetric bases of exponential functions in the closure of their linear span in the spaces Lp(μβ) and ℓp(πβ) where αμβ=tβαt, t<0, and νβ({K})=(K+1)β, K∈{0,1,2,...}. We also consider the dual problem of free interpolation in the corresponding Banach Lp-spaces of analytic functions. In many cases one gives the complete description of bases of exponential functions (in the closure of its linear span) in the above-indicated spaces and the solution of the corresponding problems of free interpolation by analytic functions.

One proves theorems on the interpolation of germs of analytic functions, defined in the neighborhoods of the interpolation nodes, in the Hardy spaces HP(0 < p ? +8), generalizing the corresponding results of N. K. Nikol'skii and V. I. Vasyunin for the classes H8 and H2. One obtains estimates of the norms of the interpolating functions in terms of the parameter of the set on which the interpolation is performed.

In this paper simple proofs are given for several propositions about continuity of singular integral operators with Cauchy kernel. Some of these propositions turn out to be consequences of more general tests for continuity of operators of the form under the condition that is a continuous operator (in a given pair of spaces). As the functions a and h one considers, as a rule, functions of the form l/(eit —eis) and respectively.

The first part of the survey is, basically, devoted to the description of the set of all the restrictions of the analytic functions of some class X onto a fixed subset E of their domain of definition; the problem consists in finding those sets E for which the corresponding class of restrictions is sufficiently simple. A tyical example is furnished by Carleson's interpolation theorem (X=H8,E is a discrete subset of the open unit circle satisfying the well-known Newman-Carleson condition; in this case the class of restrictions coincides with the set of all functions bounded on E), to which the largest part of the paper is devoted. We also present the results of L. Carleson and P. Beurling on the linear operator of the interpolation of bounded functions by bounded analytic functions and on the estimate of the uniform norm of the interpolating functions.

One constructs a new linear bounded operator which solves the problem of free interpolation in the Hardy space H8. This operator is analogous to the well-known interpolation operator of P. W. Jones. One of the fundamental distinctions of the operator constructed in this paper consists in the fact that it acts in the smallest (in a certain sense) linear closed subspace of the space H8 which, in turn, gives the possibility to obtain new results on free interpolation for several subclasses of the spaces H8. In the second part of the paper one proves a theorem which, in a significant number of cases, allows us to reduce the solution of the problem on multiple interpolation (when the multiplicities are bounded in their totality) to the problem of free interpolation with simple nodes. One gives several examples of the application of this theorem.

The second part of the survey (the first part has been published in Vol. 47 of the same publication) is basically devoted to the description of the set of all restrictions of analytic functions from the class X onto a fixed subset E of their domain of definition. Contents: the proof of Earl's theorem on interpolation by Blaschke products and some of their applications to interpolation on subclasses of the space H8; interpolation in the spaces of all functions analytic in the unit circle with a sequence of coefficients from lp(1?p?8); interpolation in some Banach spaces consisting of functions which are analytic in the open unit circle and continuous in its closure (in this case the results are similar to the well-known Rudin-Carleson theorem). In conclusion, one considers problems of interpolation in spaces of analytic functions which are smooth up to the boundary.

One considers analogues of Carleson's classical imbedding theorems for spaces of slowly increasing analytic and harmonic functions.

A series of conditions is given, imposed on a subset ? of the unit disk D, sufficient that the collection of all restrictions to the set ? of functions from the Bergman space be naturally isomorphic with the space lp(?).

Some assertions on free interpolation in spaces of functions analytic in the unit disk with boundary values from the Besov classes Bpo (T (1?p<+8,T is the unit circle) are formulated.

There is recounted a simplification of the new proof of the Newman-Carleson theorem on interpolation sequences, recently discovered by P. Jones.