On the Riesz Constants for Systems of Integer Translates

In this paper, one-parameter families of integer translates of the Gaussian and Lorentz functions are studied. For a Lorentz function, we obtain formulas for the coefficients of the series defining node functions and show that the limit value of node functions is given by a sample function. For systems of translates generated by the Gaussian and Lorentz functions as well as by the node functions related to them, we obtain explicit expressions for the Riesz constants and study the parameter-dependent behavior of these constants. While proving some of the results of this paper, we establish the monotonicity of a special ratio of two Jacobi theta functions, a fact which is of interest in itself.