Coefficient estimates for exponential series
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- by W. T. Sledd PDF
- Proc. Amer. Math. Soc. 45 (1974), 357-364 Request permission
Abstract:
Results are obtained which relate the size of the coefficients ${a_n}$ of an exponential series $f(x) = \Sigma _{n = 0}^\infty {a_n}{\varepsilon ^{ - {\Lambda _n}x}},x > 0,\operatorname {Re} {\Lambda _n} > 0$, to the function $f$. These results involve comparisons between weighted ${l^p}$ sums of the sequence $({a_n})$ and weighted ${L^p}$ integrals of $f$ on $[0,\infty )$.References
- J. M. Anderson, Boundary properties of analytic functions with gap power series, Quart. J. Math. Oxford Ser. (2) 21 (1970), 247–256. MR 264081, DOI 10.1093/qmath/21.2.247
- J. M. Anderson and K. G. Binmore, Coefficient estimates for lacunary power series and Dirichlet series. I, II, Proc. London Math. Soc. (3) 18 (1968), 36–48; 49–68. MR 223576, DOI 10.1112/plms/s3-18.1.36
- K. G. Binmore, Interpolation, approximation, and gap series, Proc. London Math. Soc. (3) 25 (1972), 751–768. MR 315135, DOI 10.1112/plms/s3-25.4.751
- R. P. Boas Jr., A general moment problem, Amer. J. Math. 63 (1941), 361–370. MR 3848, DOI 10.2307/2371530
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Dieter Gaier, On the coefficients and the growth of gap power series, SIAM J. Numer. Anal. 3 (1966), 248–265. MR 204653, DOI 10.1137/0703019 P. M. Owen, A generalization of Hilbert’s double series theorem, J. London Math. Soc. 5 (1930), 270-272.
- H. R. Pitt, Theorems on Fourier series and power series, Duke Math. J. 3 (1937), no. 4, 747–755. MR 1546029, DOI 10.1215/S0012-7094-37-00363-6
- Laurent Schwartz, Étude des sommes d’exponentielles réelles, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 959, Hermann & Cie, Paris, 1943 (French). MR 0014502
- W. T. Sledd, Coefficient estimates for Dirichlet series, Trans. Amer. Math. Soc. 150 (1970), 69–76. MR 268564, DOI 10.1090/S0002-9947-1970-0268564-4
- B. A. Taylor and D. L. Williams, Interpolation of $l^{q}$ sequences by $H^{p}$ functions, Proc. Amer. Math. Soc. 34 (1972), 181–186. MR 294652, DOI 10.1090/S0002-9939-1972-0294652-X E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 2nd ed., Clarendon Press, Oxford, 1948.
- Daniel Waterman, On some high indices theorems, Trans. Amer. Math. Soc. 69 (1950), 468–478. MR 39102, DOI 10.1090/S0002-9947-1950-0039102-3
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 357-364
- MSC: Primary 30A16
- DOI: https://doi.org/10.1090/S0002-9939-1974-0349970-5
- MathSciNet review: 0349970