Convolution Operators In Spaces Research Articles

Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations

In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important characteristic of the numerical method is that it is directly applied to the nonlocal equation by introducing the discrete convolution operator. Starting from the continuous Cauchy problem defined on the real line, we first construct the discrete Cauchy problem on a uniform grid of the real line. Thus the semi-discretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initial-value problem for this system is well-posed. We prove that solutions of the discrete problem converge uniformly to those of the continuous one as the mesh size goes to zero and that they are second-order convergent in space. We then consider a truncation of the infinite domain to a finite one. We prove that the solution of the truncated problem approximates the solution of the continuous problem when the truncated domain is sufficiently large. Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions for various convolution kernels.

Linear continuous right inverse to convolution operator in spaces of holomorphic functions of polynomial growth

We consider the convolution operator in spaces of holomorphic functions, defined in convex subdomains of the complex plane, with polynomial growth at a boundary. We prove that if this operator is surjective on the class of all bounded convex domains, then it always has a linear continuous right inverse one.

On the continuous linear right inverse for convolution operators in spaces of ultradifferentiable functions

A surjective convolution operator is considered in spaces of complex-valued Beurling ultradifferentiable functions of normal type on a finite interval. A complete description of characteristic functions for which this operator has a continuous linear right inverse is obtained. The finiteand infinite-order differential operators with constant coefficients are studied as a special case.

Convolution operators in A−∞ for convex domains

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ n and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

Spherical convolution operators in spaces of variable Hölder order

In this paper, we study the images of operators of the type of spherical potential of complex order and of spherical convolutions with kernels depending on the inner product and having a spherical harmonic multiplier with a given asymptotics at infinity. Using theorems on the action of these operators in Holder-variable spaces, we construct isomorphisms of these spaces. In Holder spaces of variable order, we study the action of spherical potentials with singularities at the poles of the sphere. Using stereographic projection, we obtain similar isomorphisms of Ho lder-variable spaces with respect to n-dimensional Euclidean space (in the case of its one-point compactification) with some power weights.

On surjectivity of the convolution operator in spaces of analytic functions

On surjectivity of the convolution operator in spaces of analytic functions

Right inverse for a convolution operator in space of germs of analytic functions on connected subsets of

Several general results concerning the existence of a continuous linear right inverse (CLRI) of a continuous linear operator are established. Using these results it is possible to obtain first (in a more general situation) necessary and then sufficient conditions (and in several cases, a test) for the existence of a CLRI in spaces of analytic germs on certain classes of connected sets for the convolution operator whose symbol is an entire function of order 1 and minimal type.

Perturbations of linear operators in spaces of holomorphic functions

The closure of the images of perturbations of convolution operators in spaces of holomorphic functions of several variables is studied. The properties of the homogeneous solutions of the corresponding perturbed equations are discovered.

Continuous linear right inverses for convolution operators in spaces of real analytic functions

We determine the convolution operators $T_μ := μ*$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction o

NONTRIVIAL EXPANSIONS OF ZERO IN ABSOLUTELY REPRESENTING SYSTEMS. APPLICATIONS TO CONVOLUTION OPERATORS

By using a general representation of nontrivial expansions of zero in absolutely representing systems of the form , where , is the Mittag-Leffler function, and are complex numbers, the author obtains a number of results in the theory of -convolution operators in spaces of functions that are analytic in -convex domains (a description of the general solution of a homogeneous -convolution equation and of systems of such equations, a topological description of the kernel of a -convolution operator, the construction of principal solutions, and a criterion for factorization).

A right inverse operator of the convolution operator

We give a description of a linear continuous right inverse operator of a convolution operator in spaces of analytic functions with an exponential basis.