TOPICS
Search

Beurling's Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
BeurlingsFunction

The entire function

B(z)=[(sin(piz))/pi]^2[2/z+sum_(n=0)^(infty)1/((z-n)^2)-sum_(n=1)^(infty)1/((z+n)^2)]
(1)
=1-(2sin^2(piz))/(pi^2z^2)[z^2psi_1(z)-z-1],
(2)

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

int_(-infty)^infty[B(x)-sgn(x)]dx=1.
(3)

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

Explore with Wolfram|Alpha

References

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.

Referenced on Wolfram|Alpha

Beurling's Function

Cite this as:

Weisstein, Eric W. "Beurling's Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BeurlingsFunction.html

Subject classifications