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 $T_a$, acting in the Segal--Bargmann--Fock space, is diagonal with respect to the canonical orthonormal basis consisting of normalized monomials. We denote by $\gamma_a$ the corresponding eigenvalues sequence. Given an arbitrary convergent sequence, we uniformly approximate it by sequences of the form $\gamma_a$ with any desired precision. We give a simple recipe for constructing $a$ in terms of Laguerre polynomials. Previously, we proved this approximation result with nonconstructive tools (Esmeral and Maximenko, ``Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences'', Complex Anal. Oper. Theory 10, 2016). In the present paper, we also include some properties of the sequences $\gamma_a$ and some properties of bounded sequences, uniformly continuous with respect to the sqrt-distance on natural numbers.