Abstract
In [1] it was shown that if a function f(z), analytic inside the unit disk, is representable by a series
and if the coefficients
rapidly tend to zero, then f(z) satisfies some functional equation ML(f) = 0. In the present paper the converse problem is solved. It is shown that if f(z) satisfies the equation ML(f)=0, then the expansion coefficients
rapidly tend to zero.
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T. A. Leont'eva, “The possible rate of decrease of the coefficients in the expansion of functions into rational fractions,” Vestn. Mosk. Univ., Mat., Mekh., No. 4, 47–55 (1973).
T. A. Leont'eva, “Representation of analytic functions by series of rational functions,” Mat. Zametki,2, No. 4, 347–356 (1967).
T. A. Leont'eva, “On the representation of functions in the unit disk by series of rational fractions,” Mat. Sb.,84(126), No. 2, 313–326 (1971).
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Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 627–639, May, 1977.
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Leont'eva, T.A. Series of rational fractions with rapidly decreasing coefficients. Mathematical Notes of the Academy of Sciences of the USSR 21, 353–360 (1977). https://doi.org/10.1007/BF01788231
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DOI: https://doi.org/10.1007/BF01788231