On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.     

On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition

We consider the spectral problem for the Schrödinger operator with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have multiple eigenvalues and a system of eigenfunctions forming a Riesz basis in L ₂(0, 1). We show that the basis property of systems of root functions of the problem can change under arbitrarily small changes in the kernel of the integral perturbation.     

On the basis property of the root functions of differential operators with matrix coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.     

The free interpolation problem and basis property of a system of rational functions in weighted Hardy spaces

The paper considers the free interpolation problem in the Hardy weighted space H p(ρ). It is assumed that the weight ρ has unique singularity of the order αonly at the point 1. Particularly, for α − p[(α + 1)p ⁻¹] > 0 the corresponding free interpolation problem is stated and its solvability is proved.     

On the Riesz basis property of the root functions in certain regular boundary value problems

The differential operatorly=y+q(x)y with periodic (antiperiodic) boundary conditions that are not strongly regular is studied. It is assumed thatq(x) is a complex-valued function of classC ⁽⁴⁾[0, 1] andq(0)q(1). We prove that the system of root functions of this operator forms a Riesz basis in the spaceL ₂(0, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 558–563, October, 1998.     

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L _p .     

Unconditional basis property of exponentials with a gap in the spectrum

Translated from Matematicheskii Zametki, Vol. 52, No. 4, pp. 62–67, October, 1992.     

On an analog of the Riesz theorem and the basis property of the system of root functions of a differential operator in L p : II

We consider an ordinary differential operator of arbitrary order, obtain necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of the system of root functions of the given operator in L _p .     

On the basis property of a trigonometric system arising in the Frankl problem

We consider the function system {cos4nθ} _n=0 ^∞ , {sin(4n ? 1)θ} _n=1 ^∞ , which arises in the Frankl problem in the theory of elliptic-hyperbolic equations. We show that this system is a Riesz basis in the space L ₂(0, π/2) and construct the biorthogonal system.     

Oblique derivative problem for nonlinear elliptic discontinuous operators in the plane with quadratic growth

Our aim is to study the oblique derivative problem for a class of nonlinear differential operators in the plane with quadratic growth. We assume the discontinuous operators to satisfy Carathéodory's condition and a suitable ellipticity condition. Under some geometrical conditions we prove strong solvability of the problem under consideration. The main tool in the proof is Leray-Schauder fixed point theorem, that reduces the solvability of the problem to the establishment of a priori estimate, by means of a step by step procedure.     

Correlation decay and the absence of zeros property of partition functions

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok for designing deterministic approximation algorithms for various graph counting and related problems. An earlier method used for the same problem is one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply often coincide or nearly coincide. In this article we show that this is not a coincidence. We establish that if the interpolation method is valid for a family of graphs, then this family exhibits a form of the correlation decay property which is asymptotic strong spatial mixing at superlogarithmic distances. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the design of the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs. Indeed this conjecture was recently confirmed by Regts. See the body of the article for details.