Beurling's Function


The entire function


where psi_1(z) is a polygamma function.

It satisfies B(z)=O(e^(2pi(I[z]))) and B(x)>=sgn(x) for all real x. Amazingly, it also has the integral


Furthermore, among all functions with the first two properties, B(x) minimizes the integral (3) (Beurling 1938, Montgomery 2001).

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Beurling, A. "Sur les intégrales de Fourier absolument convergentes et leur application à fonctionelle." Neuvième congrès des mathématiciens scandinaves. Helsingfors, 1938.Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.

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Beurling's Function

Cite this as:

Weisstein, Eric W. "Beurling's Function." From MathWorld--A Wolfram Web Resource.

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