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.
Keywords
Functional Equation Unit Disk Rational Fraction Converse Problem
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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).Google Scholar
2.
T. A. Leont'eva, “Representation of analytic functions by series of rational functions,” Mat. Zametki,2, No. 4, 347–356 (1967).Google Scholar
3.
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).Google Scholar