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Uniform Estimates of Entire Functions by Logarithmic Sums

✍ Scribed by Henrik L. Pedersen

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 442 KB
Volume: 146
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1996.3046

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✦ Synopsis

We give uniform estimates in the whole complex plane of entire functions of exponential type less than a certain numerical constant (approximately equal to 0.44) having sufficiently small logarithmic sums. In these estimates the entire dependence on the function is through its type and logarithmic sum. This result extends a theorem of Koosis about polynomials and gives a new proof of that theorem. The proof is based on material related to multiplier theorems, first obtained by Beurling and Malliavin. 1997 Academic Press Contents 1. Introduction. 2. Preliminary results. 3. Construction of pre-multipliers. 4. Lower bounds. 5. Energy. 6. Comparison of sums and integrals. 7. Functions of exponential type having finite logarithmic sums. 8. Uniform estimates by logarithmic sums. 9. Weighted approximation on the integers. 10. Appendix. A Computation.

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