Problem Of Spectral Synthesis Research Articles

О плотности многочленов в специальном пространстве целых функций экспоненциального типа

The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.

Spectral synthesis for the space of tempered solutions of a convolution system on discrete Abelian groups

We consider a problem of spectral synthesis in the topological vector space $${\mathcal{M}(G)}$$ of tempered functions on a discrete Abelian group G. It is proved that the space of tempered solutions of a convolution system on discrete Abelian groups admits spectral synthesis, that is the space of tempered solutions of a convolution system coincides with the closed linear span in $${\mathcal{M}(G)}$$ of all exponential monomial solutions of this system.

Spectral synthesis on the group of conformal automorphisms of the unit disc

Given a group of conformal automorphisms of the unit disc , the problem of spectral synthesis is solved for subspaces that are invariant under right -shifts and conjugations by elements of the subgroup . A new theorem on spectral synthesis for subspaces of that are invariant under weighted conformal shifts is an intermediate result. Bibliography: 35 titles.

Spectral synthesis for the differentiation operator in the Schwartz space

We consider the spectral synthesis problem for the differentiation operator D=d/dt in the Schwartz space E(a;b) and the dual problem of local description for closed submodules in a special module of entire functions.

Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated.A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions.

Spectral synthesis in de Branges spaces

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces \({\mathcal{H}(E)}\) . Namely, we describe the de Branges spaces \({\mathcal{H}(E)}\) such that every complete and minimal system of reproducing kernels \({\{k_\lambda\}_{\lambda \in \Lambda}}\) with complete biorthogonal \({\{g_\lambda\}_{\lambda \in \Lambda}}\) admits the spectral synthesis, i.e., \({f \in \overline{\rm Span} \{(f, g_\lambda) k_\lambda : \lambda \in \Lambda\}}\) for any f in \({\mathcal{H}(E)}\) . Surprisingly, this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, while special cases of de Branges spaces of the second class appeared in the literature only recently; we give a complete characterisation of this second class in terms of the spectral data for \({\mathcal{H}(E)}\) .

Hereditary completeness for systems of exponentials and reproducing kernels

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials {eiλnt}n∈N in L2(−a,a) is hereditarily complete up to a one-dimensional defect. This means that for any partition N=N1∪N2 of the index set, the orthogonal complement to the system {eiλnt}n∈N1∪{en′}n∈N2, where {en′} is the system biorthogonal to {eiλnt}, is at most one-dimensional. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.

An attempt to interpret the Weil explicit formula from Beurling's spectral theory

A. Beurling introduced the celebrated problem of spectral synthesis. Roughly speaking, it is a problem whether functions belonging to a certain Banach space have a possibility to be approximated by trigonometric polynomials in the appropriate topology. For this problem Beurling introduced the concept of spectral sets whose elements are regarded as exponents of trigonometric polynomials. In the Weil explicit formula we can see a certain phenomenon which may be related to Beurling's spectral sets. The purpose of this paper is to study the phenomenon.

A non-commutative Sobolev estimate and its application to spectral synthesis

In [M. K. Vemuri, Realizations of the canonical representation, Proc. Indian Acad. Sci. Math. Sci., 118 (2008), 115 –131], it was shown that the spectral synthesis problem for the Alpha transform is closely related to the problem of classifying realizations of the canonical representation (of the Heisenberg group). In this paper, it is shown that discrete sets are sets of spectral synthesis for the Alpha transform.

The failure of spectral synthesis on some types of discrete Abelian groups

The problem of spectral synthesis on arbitrary Abelian groups is solved in the negative.

Local description of closed submodules of a special module of entire functions of exponential type

Let be a system of polynomials of the complex variable . In connection with the problem of spectral synthesis for systems of differential operators , , the problem of the local description of closed submodules is considered for a special module of entire functions over the ring . It is shown that this problem can be reduced to the local description over the ring , where is the Luroth polynomial associated with the system .

Approximate spectral synthesis in the Bergman space

algebra of bounded holomorphic functions in the unit disk) can be approximated by finite Blaschke products, in the topology of uniform convergence on compact subsets of the disk. Along the way, we find a theorem about kernel functions for weighted Bergman spaces of the type that estimate them away from the diagonal. The motivation forthese results is the study of z-invariant subspaces in the Bergman space. It is known that the lattice of z-invariant subspaces in Bergman spaces has a very complicated structure. But one may single out among all z-invariant subspaces those of simplest nature, the zero-based ones. What z-invariant subspaces can be approximated by zero-based ones? What z ∗ -invariant subspaces can be approximated by finite-dimensional ones? In this paper, we answer these questions (see Theorems 1 and 2), and we discuss some relations with rational approximation and cyclic vectors forthe backward shift. Let X be a Banach space of functions analytic in the unit disk D ={ z ∈ C :| z| < 1} of the complex plane C. Suppose that X is invariant with respect to the operator Mz of multiplication by the independent variable. Many important problems concerning the structure of the lattice of subspaces of X that are closed and invariant with respect to Mz (or, simply, z-invariant) are related to problems of spectral synthesis. Let Y be the space dual to X. Then the functionals kλ of evaluation of functions in X at the points of D, kλ : f → f (λ); λ ∈ D, are eigenvectors of the operator M ∗ : � λI − M ∗ z � kλ = 0;

Spectral synthesis and local description for several variables

The problem of spectral synthesis for subspaces of analytic functions invariant under the operators of partial differentiation is reduced to the problem of the local description of closed submodules in the module of entire functions of exponential type over the ring of polynomials.

Spectral synthesis for systems of differential operators with constant coefficients. Duality theorem

Most known papers on spectral synthesis in complex domains are based on transitions from problems of spectral synthesis to equivalent problems of local description. As a rule, such a transition is carried out in the framework of some special assumptions and meets considerable difficulties. A general method developed in this paper enables one to verify the duality theorem in the setting of several complex variables, when each operator , , acts with respect to a single variable. These assumptions cover the case of a system of partial differential operators. The duality transition here breaks into three separate steps. Two of them are connected with classical results of the theory of analytic functions and only one relates to general duality theory. This allows one to speak about singling out the analytic component of the transition from a spectral synthesis problem to an equivalent problem of local description.

BMO and Lipschitz approximation by solutions of elliptic equations

We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak * spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.

SPECTRAL SYNTHESIS IN A COMPLEX DOMAIN FOR A DIFFERENTIAL OPERATOR WITH CONSTANT COEFFICIENTS. I: A DUALITY THEOREM

The problem of spectral synthesis in a complex domain for a differential operator with symbol , is reduced to the problem of a local description of the closed submodules of a module (of entire functions of exponential type) over the ring of polynomials of the form , .

SPECTRAL SYNTHESIS IN A COMPLEX DOMAIN FOR A DIFFERENTIAL OPERATOR WITH CONSTANT COEFFICIENTS. III: AMPLE SUBMODULES

In [1] and [2] the problem of spectral synthesis for a subspace W invariant under a differential operator with constant coefficients was reduced to a check of two properties of its annihilator submodule I = An W: stability and saturation. The present article describes a situation in which this check leads to a positive result, i.e., the annihilator submodule I is ample. Bibliography: 14 titles.

SPECTRAL SYNTHESIS IN A COMPLEX DOMAIN FOR A DIFFERENTIAL OPERATOR WITH CONSTANT COEFFICIENTS. II: THE MODULE METHOD

The problem of spectral synthesis for a subspace W invariant with respect to a differential operator with constant coefficients was reduced earlier to verification that its annihilator submodule I = An W is ample. In the present article the property of ampleness is split into two parts - stability, and the property of being saturated. The latter properties are subjected to a systematic investigation.

Invariant subspaces in the polydisk

This note is a study of unitary equivalence of invariant subspaces of H2 of the polydisk. By definition, this means joint unitary equivalence of the shift operators restricted to the invariant subspaces. In one variable, all invariant subspaces are unitarily equivalent and all can be represented as inner functions times H2. In several variables, our results suggest that unitary equivalence and multiplication by inner functions are again related. For example, all invariant subspaces of a given invariant subspace Jt which are unitarily equivalent to Jί are φJΐ, for φ inner; and all invariant subspaces unitarily equivalent to an invariant subspace Ji of finite codimension are yJί. In particular, two invariant subspaces of finite codimension are unitarily equivalent if and only if they are equal. 0. The classic paper of Beurling [4] led to a spate of research in operator theory, Hp theory, and other areas, which continues to the present. Although Beurling's answer to the problem of spectral synthesis was negative, his explicit characterization of the invariant subspaces of the unilateral shift, in terms of the inner-outer factorization of analytic functions has had a major impact. Since the Hardy space for the unit disk is the primary nontrivial example for so many different areas, it is not surprizing that this characterization has proved so important. Almost everyone who has thought about this topic must have considered the corresponding problem for H2 of the polydisk. Although the existence of inner functions in this context is obvious (in contrast with the case of the ball), one quickly sees that a Beurling-like characterization is not possible. Most results on this problem have gone unpublished [10], except for the book of Rudin [11], and many results are counterexamples. One exception is a characterization, by Ahern and the second author, of invariant subspaces having finite codimension, as the closure of the ideals in C[zv... ,zN] of finite codimension; [2]. In this note, we eschew the goal of characterizin g all invariant subspaces in the polydisk and instead investigate equivalence of invariant subspaces in N variables, under joint unitary equivalence of the restrictions of the N coordinate shifts. Put another way, we shall consider the problem of characterizin g the (inequivalent) submodules of the Hardy space over the polydisk algebra. The existence of nonunitarily equivalent submodules is directly related to the

On the spectral synthesis problem for points in the dual of a nilpotent Lie group

On the spectral synthesis problem for points in the dual of a nilpotent Lie group