Bessel Polynomial Expansions in Spaces of Holomorphic Functions

General methods of nuclear Fr´echet spaces in conjunction with the theory of K¨othe sequence spaces have been used to obtain a basis criterion for power sequence spaces. How this basis criterion is thus yielding a refinement of Cannon’s criterion in the case of space of holomorphic functions is illustrated here. Hence, we give a characterization of those infinite matrices for which the respective Bessel polynomials constitute a basis in the space of holomorphic functions. Conse- quently, an answer to the question posed by H. L. Krall and O. Frink Trans. Amer. Math. Soc. 65, 1949, 100]115. concerning the actual expansion of a holomorphic function in terms of Bessel polynomials is given. Also, we append a new criterion for basis transforms in such a space of holomorphic functions by means of order of magnitude of coefficients of polynomials.