Approximation of Entire Functions in Some Banach Spaces and Its Relation to Growth Function

This paper delves into the intricate relationship between the generalized order of an entire function, defined by a model function, and the rate at which this function can be approximated by polynomials within the unit disk. Our investigation unveils novel connections, establishing a precise relationship between the generalized order of the entire function and the convergence of its best polynomial approximations. The present findings significantly expand upon earlier seminal works by A.R. Reddy, I.I. Ibragimov and N.I. Shyhaliev, S.B. Vakarchyk, and numerous other researchers in the field.