Victor D Didenko

• Dr. Sc.
• Professor at Southern University of Science and Technology

Necessary conditions for the invertibility of Toeplitz plus Hankel operators T(a) + H(b) with generating functions a, b ∈ L∞(T(double-struck)) satisfying the relation a(t)a(1/t) = b(t)b(1/t), t ∈ T(double-struck) are obtained. In addition, sufficient conditions for the invertibility of such operators are also provided and efficient representations...

Considered is the equation $$ (T(a)+H(b))\phi=f, $$ where $T(a)$ and $H(b)$, $a,b\in L^\infty(\mathbb{T})$ are, respectively, Toeplitz and Hankel operators acting on the classical Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the generating functions $a$ and $b$ satisfy the so-called matching condition [1,2], $$ a(t) a(1/t)=b(t)b(1/t), \, t\in \...

An explicit index formula for Toeplitz plus Hankel operators with piecewise quasicontinuous generating functions is obtained. Moreover, for some classes of the operators mentioned conditions of one-sided invertibility are established.

Various aspects of numerical analysis for equations arising in boundary integral equation methods have been the subject of several books published in the last 15 years [95, 102, 183, 196, 198]. Prominent examples include various classes of o- dimensional singular integral equations or equations related to single and double layer potentials. Usually...

Toeplitz plus Hankel operators T(a)+H(b), a,b\in L^\infty acting on the classical Hardy spaces H^p, 1<p<\infty, are studied. If the generating functions $a$ and $b$ satisfy the so-called matching condition a(t) a(1/t)=b(t) b(1/t), an effective description of the structure of the kernel and cokernel of the corresponding operator is given. The result...

The invertibility of Toeplitz plus Hankel operators \(T(a)+H(b)\), \(a,b\in L^\infty \) acting on \(l^p\)-spaces is studied. If the generating functions a and b satisfy the equation various sufficient conditions for the invertibility and one-sided invertibility of the operators \(T(a)+H(b)\) are obtained and the corresponding inverses are construct...

The invertibility of Toeplitz plus Hankel operators $T(\mathcal{A})+H(\mathcal{B})$, $\mathcal{A},\mathcal{B}\in L^\infty_{d\times d}(\mathbb{T})$ acting on vector Hardy spaces $H^p_d(\mathbb{T})$, $1<p<\infty$, is studied. Assuming that the generating matrix functions $\mathcal{A}$ and $\mathcal{B}$ satisfy the equation \begin{equation*} \mathcal{...

The invertibility of Wiener–Hopf plus Hankel operators W(a)+H(b) acting on the spaces L^p(\mathbb{R}^+) , 1 \leq p<\infty is studied. If a and b belong to a subalgebra of L^\infty(\mathbb{R}) and satisfy the condition a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, we establish necessary and also sufficient conditions for the operators W(a)+H(b) to be...

The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and $l^p$-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. Special attention is paid to a newly developed method, which allows to establish necessary, sufficient and also necessary a...

The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition \begin{equation*} a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, \end{equation*} we establish necessary and also sufficient con...

Toeplitz plus Hankel operators $T(a)+H(b)$ acting on Hardy spaces $H^p(\sT)$, $p\in(1,\infty)$, where $\sT$ is the unit circle, are studied. If the functions $a,b\in L_\infty(\mathbb{T})$ satisfy the relation $a(t)a(1/t)=b(t)b(1/t)$, $t\in\mathbb{T}$ and the Toeplitz operators $T(ab^{-1})$ and $T(a\widetilde{b}^{-1})$, $\widetilde{b}(t)=b(1/t)$ hav...

Toeplitz T(a) and Toeplitz plus Hankel operators T(a) + H(b) acting on sequence space lp , 1 < p < ∞, are considered. If a ∈ PCp is a piecewise continuous lp -multiplier, a complete description of the kernel of the Fredholm operator T(a) is derived. Moreover, the kernels of Fredholm Toeplitz plus Hankel operators T(a) + H(b) the generating function...

Toeplitz T(a) and Toeplitz plus Hankel operators T(a) + H(b) acting on sequence space lp, 1 < p < ∞, are considered. If a ε PCp is a piecewise continuous lp-multiplier, a complete description of the kernel of the Fredholm operator T(a) is derived. Moreover, the kernels of Fredholm Toeplitz plus Hankel operators T(a)+H(b) the generating functions a...

The Nyström method for double layer potential equations on simple closed piecewise smooth contours is stable if and only if certain operators associated with the opening angles of the corners are invertible. Numerical experiments show that there are opening angles which cause the instability of the method.

We find the “norm” of a power function in the Gurov–Reshetnyak class on the real line. Moreover, as a result of numerical experiments, we establish a lower bound for the norm of the operator of even extension of a function from the Gurov–Reshetnyak class from the semiaxis onto the entire real line.

Considered are Wiener–Hopf plus Hankel operators \(W(a)+H(b)\;:\;L^{p}(\mathbb{R}^{+})\rightarrow L^{p}(\mathbb{R}^{+})\) with generating functions a and b from a subalgebra of \(L^{\infty}(\mathbb{R})\) containing almost periodic functions and Fourier images of \(L^{1}(\mathbb{R})-\) functions. If the generating functions a and b satisfy the match...

Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators Rτ associated with the corner points τ. The operators Rτ do not depend on the shape of the contour but only on the opening angles of the corner points τ. The invertibility o...

A "norm" of power function in the Gurov-Reshetnyak class on the real line is computed. Moreover, a lower bound for the norm of the operator of even extension from the semi-axis to the whole real line in the Gurov-Reshetnyak class is obtained from numerical experiments.

Generalized Toeplitz plus Hankel operators $T(a)+H_{\alpha}(b)$ generated by functions $a,b$ and a linear fractional Carleman shift $\alpha$ changing the orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$ and satisfy the condition $$ a(t) a(\...

Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators $R_{\tau}$ associated with the corner points $\tau$. The operators $R_{\tau}$ do not depend on the shape of the contour but only on the opening angles of the corner points $...

Considered are Wiener--Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions. If the generating functions $a$ and $b$ satisfy the matching condition \begin{e...

Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corres...

Considered is the equation \((T(a)+H(b))\phi =f\), where T(a) and H(b), \(a,b\in L^\infty (\mathbb {T})\) are, respectively, Toeplitz and Hankel operators acting on the classical Hardy space \(H^p(\mathbb {T})\), \(1<p<\infty \). If the generating functions a and b satisfy the matching condition \(a(t) a(1/t)=b(t)b(1/t), \, t\in \mathbb {T}\), two...

Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corres...

Spline Galerkin approximation methods for the Sherman-Lauricella integral equation on simple closed piecewise smooth contours are studied, and necessary and sufficient conditions for their stability are obtained. It is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the conto...

The stability of the Nystroem method for the double layer potential equation on simple closed piecewise smooth contours is studied. Necessary and sufficient conditions of the stability of the method are established. It is shown that the method under consideration is stable if and only if certain operators associated with the opening angles of the c...

A spectral problem for the Sturm–Liouville equation on the edges of an equilateral regular star-tree with the Dirichlet boundary conditions at the pendant vertices and Kirchhoff and continuity conditions at the interior vertices is considered. The potential in the Sturm–Liouville equation is a real–valued square summable function, symmetrically dis...

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t)=b(t)b(1/t). Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions o...

Wiener-Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions are studied. For $a$ and $b$ satisfying the so-called matching condition \begin{equation*} a(t)...

Regular stationary stochastic vector processes whose spectral densities are the boundary values of matrix functions with bounded Nevanlinna characteristic are considered. A criterion for the representability of such processes as output data of linear time invariant dynamical systems is established.

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very...

The main property of functions with bounded mean oscillations-viz. the exponential decay of the distribution function, is considered. This property is represented by the John–Nirenberg inequality but the exact constant in the exponent is known only for the functions defined on those intervals which are at least one-sided bounded. In the present pap...

The stability of the Nystr\"om method for the Muskhelishvili equation on piecewise smooth simple contours $\Gamma$ is studied. It is shown that in the space $L^2$ the method is stable if and only if certain operators $A_{\tau_j}$ from an algebra of Toeplitz operators are invertible. The operators $A_{\tau_j}$ depend on the parameters of the equatio...

For a function φ non-negative on the interval [0, 1], the power mean of order α ≠ 0 is defined by the equality $ \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 $. We consider the class ${\widetilde{{RH}}^{\alpha, \beta }}(B)$ of functions φ satisfying the reverse...

This paper is devoted to the study of $L_2$-solutions of the operator equations \begin{equation}\label{abst1} f-B_M \mathfrak{F} a \mathfrak{F}^{-1} f= g \end{equation} where $a$ is the operator of multiplication by a matrix $a\in L_\infty^{m \times m}(\sR^s)$, $m,s\in\sN$, $\mathfrak{F}$ denotes the Fourier transform, and $B_M$ is the dilation ope...

The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points c j , j=0,1,⋯,m, relies on the invertibility of certain operators A c j belonging to an algebra of Toeplitz operators. The operators A c j do not depend on the shape of the contour, but on the opening angle θ j of the corresponding corner c j and...

The stability of the Nystroem method for the Sherman–Lauricella equation on piecewise smooth closed simple contour Γ is studied. It is shown that in the space L2 the method is stable if and only if certain operators associated with the corner points of Γ are invertible. If Γ does not have corner points, the method is always stable. Numerical experi...

Let $w$ be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse H\"older inequality with exponents $0<\alpha<\beta$. In the present paper, sharp estimates of the compositions of the power means $\mathcal{P}_\alpha w(x):=\left ((1/x)\int_0^x w^\alpha(t)\,dt\right )^{1/\alpha}$, $x>0$, are obtained for vario...

Let w be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents 0 < α < β. In the present paper, sharp estimates of the compositions of the power means P α w(x) := ((1/x) ∫ x 0 w α (t) dt) 1/α , x > 0, are obtained for various exponents α. As a result, for the function w a pr...

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space $L_2^m(\sR^s)$, $m\geq1$ and $s\geq1$, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix $M$ and the matrix function $c$ defining the corresponding refinem...

Kernels of functional operators generated by mapping that possess complete quasi-wandering sets are studied. It is shown that the kernels of the operators under consideration either consist of a zero element or contain a subset isomorphic to a space \( L_\infty \left( \mathbb{S} \right) \)), where \( \mathbb{S} \subset \mathbb{R}^n \) has a positiv...

The solvability and Fredholm properties of refinement equations in spaces of square-integrable functions are studied. Necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and non-homogeneous refinement equations are established. It is shown that in the space $L_2(\sR)$ the kernelspace of any homogeneous equat...

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theore...

The Galerkin and collocation methods were implemented on several examples, using splines of order m = d + 1. We also performed an extensive investigation on the performance of the proposed schemes, examining in particular the behaviour of the code as a function of the various available parameters.

Approximation methods for the Riemann-Hilbert boundary problem are considered in this chapter. A particular feature of these problems is that the operators studied act in a pair of spaces, so the corresponding operator spaces do not have any multiplication operation which makes the use of algebraic techniques more difficult. However, by introducing...

In this chapter, we study different problems connected with approximation methods for singular integral equations of the form $$ \begin{gathered} t) \equiv a(t)u(t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{u(s) ds}} {{s - t}}} + c(t)\overline {u(t)} + \frac{{d(t)}} {{\pi i}}\int_\Gamma {\frac{{\overline {u(s)} ds}} {{s - t}} + \frac{{e(t)}} {{\...

We start with polynomial and spline approximation methods for the Cauchy singular integral equation $$ \begin{gathered} (Au)(t) \equiv a(t)\phi (t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{\phi (\tau ) d\tau }} {{\tau - t}}} + \overline {c(t)\phi (t)} + \overline {\frac{{d(t)}} {{\pi i}}\int_\Gamma {\frac{{\phi (\tau ) d\tau }} {{\tau - t}}} }...

A complex algebra is an associative ring A which is also a complex vector space. It is assumed that vector space addition and ring addition coincide and that the operations of multiplication and multiplication by scalars satisfy the relation $$ \lambda (xy) = (\lambda x)y = x(\lambda y) $$ (1.1) for all x, y ∈ A and for all complex numbers λ.

Let D ⊂ ℝ2 denote a domain bounded by a simple closed piecewise smooth contour Γ and let \( {\bar D} \) := D ∪ Γ. It is well known that many problems in plane elasticity, radar imaging, and theory of slow viscous flows can be reduced to the biharmonic problem $$ \Delta ^2 U(x,y) = 0, (x,y) \in D, $$ where Δ is the Laplace operator (3.48). We assume...

The solvability of multivariate refinement equations in the Hilbert space L2(ℝs) is studied. It is shown that the set of all L2-solutions of such equations either consists of the trivial solution only or it contains a subspace isomorphic to a space L∞(SM) where SM is a subset of ℝs with a positive Lebesgue measure. Therefore, the corresponding mult...

Approximate foveated images can be obtained from uniform images via the approximation of some integral operators. In this paper it is shown that these operators belong to a well studied operator algebra, and the problem of restoration of the approximate uniform pre-images is considered. Under common assumptions on smoothness of the integral operato...

We investigate the possibility of applying approximation methods to the famous Muskhelishvili equation on simple closed smooth curve $\Gamma $. Since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way. It is shown that the spline Galerkin, spline collocation and spline qualocation methods...

The paper presents lower bounds for the spectral radii of refinement and subdivision operators with continuous matrix symbols and with dilations from a class of isotropic matrices. This class contains main dilation matrices used in wavelet analysis. After obtaining general formulas, two kinds of estimates for the spectral radii are established: nam...

Using the Goursat representation for the bi-harmonic function and approximate solutions of a corrected Muskhelishvili equation we construct approximate solutions for biharmonic problems in smooth domains of ℝ 2 . It is shown that the sequence of these approximate solutions converges uniformly on each compact subset of the initial domain D. Under ad...

The spectral radii of refinement and subdivision operators considered on the space L-2 can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if T is the unit circle and if the symbol a of a refinement operator satisfies the condi...

In this paper we consider simple methods for the numerical solution of Cauchy singular integral equations with conjugation based on quadratures. They use collocation at a set of nodes which is graded toward the endpoints, to accomodate for the presence of the singularity. Our numerical experience reveals that the theoretical stability findings are...

Convolution dilation operators with non-compactly supported kernels are considered and effective formulae for their spectral radii are found. The formulae depend on the behaviour of the eigenvalues of the dilation matrix.

Let D ⊂ ℝ2 denote a domain bounded by a simple closed piecewise smooth contour Γ and let D̄ := D ∪ Γ. It is well known that many problems in plane elasticity, radar imaging, and theory of slow viscous flows can be reduced to the biharmonic problem Δ2U (x,y) = 0, (x,y) ∈ D, where Δ is the Laplace operator (3.48). We assume that the function U is fro...

A quadrature method based on locally non-equidistant meshes is proposed for the approximate solution of the biharmonic Dirichlet problem on a non-smooth domain. Necessary and sufficient conditions for stability are established.

For the approximate solution of the biharmonic Dirichlet problem we propose and study a boundary element method based on the integral equation of Muskhelishvili. Such an approach has a number of advantages, for instance, this equation does not have any critical geometry and in the case of smooth boundaries the method always converges. If the bounda...

An approximation method for a wide class of two-dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily...

In this paper, we study an approximation method for solving singular integral equations with conjugation on an open arc. The stability of the method depends on the invertibility of certain operators which belong to well-known algebras. We investigate properties of these operators and show how to choose the parameters of the approximation method so...

Approximation methods for singular integral operators with continuous coefficients and conjugation on curves with corners are investigated with respect to their stability. Particular emphasis is devoted to index constraints for the local stability conditions. It turns out that, if an associated local operator is Fredholm, then the absolute value of...

A real extension à of a complex C*-algebra A by some element m which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on Ã. After then we are able to study Moore-Penrose invertibility in Ã. Because this notion strongly depends on the element m,...

A number of numerical methods for singular integral operators with continuous coefficients on curves with corners are studied. Necessary and sufficient conditions for stability of the methods under consideration are given. These conditions are formulated in terms of invertibility of some model operators. It is shown that the model operators belong...

An extension of a by some element m, which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on after which we are able to study different kinds of Moore-Penrose invertibility in . Due to such an approach, we obtain necessary and sufficient cond...

A quadrature method for double layer potential equations with continuous coefficients on piecewise smooth curves is studied. The underlying grids are supposed to be locally non-equidistant. Such grids occur for instance in adaptive methods. Necessary and sufficient conditions for stability of the method are given in terms of invertibility of some w...

We prove some stability results for a simple numerical method for Mellin convolution operators with conjugation and investigate the Fredholm property for its local operators.

A numerical method for singular integral equations with conjugation is studied. For the first time, such kind of methods (the method of discrete vortices) was presented by S.M.Belotserkovskii in the middle of the 50-s when he was studying some problems of aerodynamics. Twenty years after I.K.Lifanov and Ya.E.Polonsky proved its stability for singul...

Approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double layer potential operator on curves with corners are investigated with respect to their stability. The methods under consideration include several kinds of quadrature rules, collocation and qualocation methods. The approach is...

We study the stability of special operator sequences. This allows us to establish necessary and sufficient conditions for the stability of a variety of approximation methods for singular integral equations with conjugation from a unique point of view.

This paper gives necessary and sufficient conditions for the applicability of collocation and Galerkin methods to bisingular integro-differential equations with continuous coefficients. They are obtained by a local principle for paraalgebras.

The approximate solution of equations with additive continuous operators is considered. Necessary and sufficient conditions are established for convergence in Lp, 1 <p < ∞,of the reduction and collocation methods applied directly to singular integral equations with conjugation.

The Riemann boundary problem is studied under the assumption that the coefficient of the problem is a complex orthogonal matrix. In this case a property of the partial indices of the problem is established together with certain properties of the canonical matrices, which are then used to construct the canonical matrix of a complex orthogonal matrix...