Constructive approximation of convergent sequences by eigenvalue sequences of radial Toeplitz–Fock operators

It is well known that for every measurable function a , essentially bounded on the positive halfline, the corresponding radial Toeplitz operator Tₐ T a , acting in the Segal–Bargmann–Fock space, is diagonal with respect to the canonical orthonormal basis consisting of normalized monomials. We denote by γ ₐ γ a the corresponding eigenvalues sequence. Given an arbitrary convergent sequence, we uniformly approximate it by sequences of the form γ ₐ γ a with any desired precision. We give a simple recipe for constructing a in terms of Laguerre polynomials. Previously, we proved this approximation result with non-constructive tools (Esmeral and Maximenko in Complex Anal. Oper. Theory 10, 2016). In the present paper, we also include some properties of the sequences γ ₐ γ a and some properties of bounded sequences, uniformly continuous with respect to the sqrt-distance on natural numbers.