Akhiezer's theorem

In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.[1]

Akhiezer's theorem

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In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.[1]

Statement[]

Let f(z) be an entire function of exponential type τ, with f(x) ≥ 0 for real x. Then the following are equivalent:

f ( z ) = F ( z ) F ( z ¯ ) ¯ {displaystyle f(z)=F(z){overline {F({overline {z}})}}}

"{displaystyle

  • One has:

| Im ( 1 / z n ) | < {displaystyle sum |operatorname {Im} (1/z_{n})|<infty }

"{displaystyle

where zn are the zeros of f.

It is not hard to show that the Fejér–Riesz theorem is a special case.[2]

Notes[]

  1. ^ see Akhiezer (1948).
  2. ^ see Boas (1954) and Boas (1944) for references.

References[]

  • Boas, Jr., Ralph Philip (1954), Entire functions, New York: Academic Press Inc., pp. 124–132

  • Boas, Jr., R. P. (1944), "Functions of exponential type. I", Duke Math. J., 11: 9–15, doi:10.1215/s0012-7094-44-01102-6, ISSN 0012-7094
  • Akhiezer, N. I. (1948), "On the theory of entire functions of finite degree", Doklady Akademii Nauk SSSR, New Series, 63: 475–478, MR 0027333


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