Appell polynomial expansions and biorthogonal expansions in Banach spaces

Buckholtz, J. D.

doi:10.1090/S0002-9947-1973-0333210-0

Appell polynomial expansions and biorthogonal expansions in Banach spaces
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by J. D. Buckholtz PDF

Trans. Amer. Math. Soc. 181 (1973), 245-272 Request permission

Abstract:

Let $\{ {p_k}\} _0^\infty$ denote the sequence of Appell polynomials generated by an analytic function $\phi$ with the property that the power series for $\theta = 1/\phi$ has a larger radius of convergence than the power series for $\phi$. The expansion and uniqueness properties of $\{ {p_k}\}$ are determined completely. In particular, it is shown that the only convergent $\{ {p_k}\}$ expansions are basic series, and that there are no nontrivial representations of 0. An underlying Banach space structure of these expansions is also studied.

References

Ralph P. Boas Jr. and R. Creighton Buck, Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 19, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin, 1964. Second printing, corrected. MR 0162914, DOI 10.1007/978-3-662-25170-6
J. D. Buckholtz, Series expansions of analytic functions, J. Math. Anal. Appl. 41 (1973), 673–684. MR 340564, DOI 10.1016/0022-247X(73)90237-0
W. T. Martin, On Expansions in Terms of a Certain General Class of Functions, Amer. J. Math. 58 (1936), no. 2, 407–420. MR 1507163, DOI 10.2307/2371049
G. A. Read, Expansions in series of polynomials, J. London Math. Soc. 43 (1968), 655–657. MR 228694, DOI 10.1112/jlms/s1-43.1.655