Beurling's Function
The entire function
where 
is a polygamma function.
It satisfies ![B(z)=O(e^(2pi(I[z])))](http://mathworld.wolfram.com/images/equations/BeurlingsFunction/Inline8.gif)
and 
for all real 
. Amazingly, it also has the integral
![int_(-infty)^infty[B(x)-sgn(x)]dx=1.](http://mathworld.wolfram.com/images/equations/BeurlingsFunction/NumberedEquation1.gif) |
(3)
|
Furthermore, among all functions with the first two properties, 
minimizes the integral (3) (Beurling 1938, Montgomery 2001).
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.
