Generalized Hardy inequalities and pseudocontinuable functions

Given positive integersn ₁<n ₂<... we show that the Hardy-type inequality $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ holds true for allf∈H ¹, provided that then _k 's, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces ofH ¹ for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.     

Boundary convergence of vector-valued pseudocontinuable functions

In the first part of the paper we discuss a multi-dimensional analogue of the well-known construction by D. Clark that allows one to study families of spectral measures of perturbations of the model contraction. In the second part we present extensions of the relevant results on the boundary behavior of pseudocontinuable functions. We show that, although the most direct analogue of the scalar theorem on the existence of boundary values for pseudocontinuable functions with respect to Clark measures fails in the non-scalar situation, suitable vector-valued versions of such results can be found.     

Inner functions in weak Besov spaces

It is shown that inner functions in weak Besov spaces are precisely the exponential Blaschke products.     

Inner functions on spaces of homogeneous type

One shows that the methods of M. Hakim, N. Sibony, and E. Løw, used by them for the construction of inner functions in a sphere, can be applied also in a more general situation. The fundamental result of the paper is: For any positive continuous function H on the unit sphere S of the space ^d, there exists a real function u, harmonic in the unit ball, such that the function u is bounded in B and ¦u¦ =H almost everywhere on S.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 7–14, 1983.     

On a maximum principle for pseudocontinuable functions

The main objective of this paper is stated in the title. In addition, a characterization of one-component inner functions is presented. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 26–35.     

On the zeros of functions from Bergman spaces and some related spaces

We obtain sharp necessary conditions on the counting function of zeros of analytic functions from Bergman spaces with standard weight and from spaces which are their natural generalization.     

On the existence of nontangential boundary values of pseudocontinuable functions

Let θ be an inner function, let θ^*(H ²)=H ²⊖θH ², and let μ be a finite Borel measure on the unit circle . Our main purpose is to prove that, if every functionf∈θ^*(H ²) can be defined μ-almost everywhere on in a certain (weak) natural sense, then every functionf∈θ^*(H ²) has finite angular boundary values μ-almost everywhere on . A similar result is true for the L^p-analog of θ^*(H ²) (p>0). Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 5–17.     

Inner product quadrature formulas exact on maximal product spaces of functions

This work extends and complements earlier work of the author [1]. An Inner Product Quadrature Formula (I P Q F) is used when approximating the definite integral of the product of two (or more) funcitons, i.e. σ_?1¹w(x) f(x) g(x) dx, where w is a weight function. The functions f and g are approximated by interpolatory functions f_φ? span φ_{ii = 0}^γ, g_ψ? span ψj_{j = 0}^δ, and the integral σ_?1¹w(x)f_φ(x)g_ψ (x) dx is evaluated exactly. Maximal values which the numbers γ and δ may take are investigated. Numerical examples of I P Q F are given. Also, the applicaitons of I P Q F in higher dimensions are commented on.     

Univalent functions in Hardy, Bergman, Bloch and related spaces

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several M?bius invariant function spaces, such as BMOA, Q _p or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space are the same as the univalent functions in H _α ^p and S _α ^p if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller M?bius invariant subspace of the Bloch space than Q _p still containing all univalent Bloch functions. This research has been supported in part by the MEC-Spain MTM2005-07347, the Spanish Thematic Network MTM2006-26627-E, and the Academy of Finland 210245.     

On the zeros of functions in Bergman spaces and in some other related classes of functions

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f?0 is a function which belongs to the Bergman space Ap (0<p<∞) and {zk} is a sequence of zeros of f which is contained in a Stolz angle, then {zk} satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.     

Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.     

Ultrametric spaces and related Hilbert spaces

We present simple proofs of the possibility of embedding ultrametric spaces in Hilbert spaces. The main part of the paper deals with ultrametric spaces that we call totally infinite spaces. Related Hilbert spaces, automorphisms of totally infinite spaces, and the corresponding linear operators are considered. Transplated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 223–237, August, 1997. Translated by V. E. Nazaikinskii     

Hermite functions and weighted spaces of generalized functions

We prove that the Hermite functions are an absolute Schauder basis for many weighted spaces of (ultra)differentiable functions and ultradistributions including the space of Fourier hyperfunctions. The coefficient spaces are also determined. Dedicated to Professor H.-G. Tillmann on the occasion of his 80th birthday     

On spaces of functions between classifying spaces

In this paper the results of Dwyer and Zabrodsky [DZ] are extended by showing that ifL is a compact Lie group andG is either ap-group or a torus, then every mapf:BG →BL is homotopic to one induced by a homomorphismφ :G →L, and two such induced maps are homotopic if and only if the corresponding homomophisms are conjugate. Several other results related to maps between classifying spaces, completions, and fibrations are also deduced.