Explicit Bernstein-Type Estimates for Fractional Hermite Functions

This paper investigates explicit coefficient-based estimates for a class of fractional Hermite functions defined through finite power series with Gamma-function coefficients. These functions may be viewed as a fractional Hermite-type family associated with the Caputo fractional derivative of order α∈(0,1]. An explicit representation of the fractional derivative is obtained as a finite sum of monomials with computable Gamma coefficients. This representation is used to derive a preliminary uniform estimate on bounded intervals [0,R] with an explicit constant depending on α, n, and R. Consistency with the integer-order setting is established by showing that, when α=1, the construction reduces to a Hermite-type polynomial family and the Caputo derivative coincides with the ordinary derivative. Explicit asymptotic formulas are obtained for the associated coefficient envelope as R→0+ and R→∞. Numerical experiments up to degree n=7 show that the ratio between the coefficient envelope and the computed supremum norm remains below approximately 1.45 for the tested parameter range. In addition, a weighted L2 estimate is derived with respect to a fractional Gaussian-type weight, yielding an explicit coefficient-based bound. The estimates obtained in this work are preliminary in nature, being based on coefficient-wise majorization, and are not claimed to be optimal. Determining sharp constants and establishing genuine norm-comparison inequalities remain open problems. The results presented here provide a rigorous starting point for the study of explicit coefficient-based estimates for fractional Hermite functions and suggest several directions for future research in fractional approximation theory.