Solvability of convolution equations in the Beurling spaces of ultradifferentiable functions of mean type on an interval

Abstract

We establish a solvability criterion for convolution equations in the Beurling classes of ultradifferentiable functions of mean type on an interval. Under study is also the question of degeneration of convolution equations into infinite-order equations with constant coefficients.

On solutions of convolution equations in spaces of ultradifferentiable functions

“…In the present paper, we establish the form of a particular solution of (1.1). Together with the results of [2], this enables us to write out the general solution of equations (1.1) and (1.2) in E 1 (ω) (I). Presently, convolution equations have been studied fairly well also in Beurling spaces of UDF of maximal type (see, e.g., [3,4,5]).…”

“…The paper is devoted to a description of solutions of the convolution equation Here T μ is the surjective convolution operator that acts linearly and boundedly on E 1 (ω) (I) by the rule (T μ f )(x) = ψ μ , f(x + • ) , f ∈ E 1 (ω) (I), x ∈ I; ψ μ is a continuous linear functional in E 1 (ω) (I) called usually the symbol of T μ . For simplicity, by the symbol of T μ we shall mean the Fourier-Laplace transform μ of ψ μ rather than this functional itself.…”

“…For simplicity, by the symbol of T μ we shall mean the Fourier-Laplace transform μ of ψ μ rather than this functional itself. So, the symbol is an entire function that is a multiplier for the weighted space of entire functions H 1 (ω),I = f ∈ H(C) ∃q ∈ (0, 1), ∃l ∈ (0, a) : f ω,q,l = sup z∈C |f (z)| e qω(z)+l| Im z| < ∞ , which is isomorphic to E 1 (ω) (I) . It is known (see [1,Theorem 2]) that, as a partial case, equations (1.1) include differential equations of infinite order with constant coefficients 1 (ω) (I).…”

“…Earlier, in [1], all symbols μ of the surjective operators T μ were described, i.e., of all convolution equations (1.1) solvable in E 1 (ω) (I) for every right-hand side g ∈ E 1 (ω) (I). Next, in [2] an exponential polynomial basis was constructed in ker T μ , i.e., in the space of solutions of the homogeneous differential equation T μ f = 0.…”

“…A particular solution for equation (1.1) will be sought in the form f = ∞ j=1 f j e −iν j x , where the exponents ν j , j ∈ N, are chosen in such a way that the system {e −iν j x : j ∈ N} be an absolutely representative system in E 1 (ω) (I) (an ARS, for short) and the |μ(ν j )|, j ∈ N, obey appropriate lower estimates. Finally, we shall prove that if ∞ j=1 g j e −iν j x is an expansion of the right-hand side g in (1.1) in an absolutely convergent series, then the function…”