## Hilbert's Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Extend Hilbert's Inequality by letting and

 (1)

so that
 (2)

Levin (1937) and Steckin (1949) showed that

 (3)

and

 (4)

Mitrinovic et al. (1991) indicate that this constant is the best possible.

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hilbert/hilbert.html

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, 1991.

Steckin, S. B. On the Degree of Best Approximation to Continuous Functions.'' Dokl. Akad. Nauk SSSR 65, 135-137, 1949.