Recursive expansions with respect to a chain of subspaces

In this work, recursive expansions in Hilbert space H = L ₂[0, 1] are considered. We discuss a related notion of frames in finite-dimensional spaces. We also suggest a constructive approach to extend an arbitrary basis to obtain a tight frame. The algorithm of extending is applied to bases of a special form, whose Gram matrix is circulant. A construction of a chain of nested subspaces { Vⁿ }_{n = 1}^￥ \left\{ {{V^n}} \right\}_{n = 1}^\infty is given, and in its foundation lies an example of a function that can be expressed as a linear combination of its contractions and translations. The main result of the paper is the theorem that provides the uniform convergence of recursive Fourier series with respect to the chain { Vⁿ }_{n = 1}^￥ \left\{ {{V^n}} \right\}_{n = 1}^\infty for continuous functions.     

On the convergence of orthorecursive expansions in nonorthogonal wavelets

The present paper is concerned with orthorecursive expansions which are generalizations of orthogonal series to families of nonorthogonal wavelets, binary contractions and integer shifts of a given function φ. It is established that, under certain not too rigid constraints on the function φ, the expansion for any function f ∈ L ²(?) converges to f in L ²(?). Such an expansion method is stable with respect to errors in the calculation of the coefficients. The results admit a generalization to the n-dimensional case.     

On orthorecursive expansions in terms of a chain of systems

Doklady Mathematics -     

A condition for almost everywhere convergence of orthorecursive expansions

An almost everywhere convergence condition with the Weyl multiplier W111111111(n) = v n is obtained for orthorecursive expansions that converge to the expanded function in L².     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

Polynomial expansions of Boolean functions with respect to nondegenerate functions

Translated from Algebra i Logika, Vol. 30, No. 6, pp. 631–637, November–December, 1991.     

Stabilization of linear dynamical systems with respect to arbitrary linear subspaces

The classical notion of stabilizing a controlled dynamical system to some specified equilibrium point is extended to include stabilization to a specified linear subspace. Necessary and sufficient conditions for existence of a solution are derived and an explicit solution recipe is given for one special case.     

A characterization of additive iterative roots of identity with respect to invariant subspaces

Aequationes mathematicae - Let $$\mathbb {K}$$ be an extension of the field  $$\mathbb {Q}$$ of rational numbers. We prove an algebraic characterization of the additive iterative roots of...     

Vector sampling expansions in shift invariant subspaces

Multi-input multi-output (MIMO) sampling scheme which is motivated by applications in multi-channel deconvolution and multi-source separation has been investigated in many aspects. Common for most of results on MIMO systems is that the input signals are supposed to be band-limited. In this paper, we study vector sampling expansions on general finitely generated shift-invariant subspaces. Necessary and sufficient conditions for a vector sampling theorem to hold are given. We also give several examples to illustrate the main result.     

Stability of line bundle transforms on curves with respect to low codimensional subspaces

We show the stability of certain syzygies of line bundles oncurves, which we call line bundle transforms. Furthermore, weprove the existence of reducible theta divisors for the transformshaving integral slope. A line bundle transform is the kernelof the evaluation map on a subspace of the space of global sections.     

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

Renormalized diagram expansions with respect to an effective parameter for charged model of water

Diagram series with respect to the electrostatic part of the interaction are constructed for the charged models widely used in the numerical modeling of polar systems. A renormalization procedure consisting of transition to irreducible diagrams is proposed. As a result of this transition, expansions with respect to an effective small parameter arise. The constructed expansion is used to solve the problem of the surface polarization at the water—vapor interface.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 97–108, April, 1995.     

On the Willems closure with respect to Ws

We determine when a submodule is Willems with respect to thespace of distributions that are tempered in the spatial direction. Received 14 October 2002. Accepted 9 February 2003.