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 $p,q>1$ and

$${1\over p}+{1\over q}\geq 1,$$ (1)

so that

$$0<\lambda= 2-{1\over p}-{1\over q}\leq 1.$$ (2)

Levin (1937) and Steckin (1949) showed that

$$\sum_{m=1}^\infty \sum_{n=1}^\infty {a_mb_n\over(m+n)^\lambd... ...p}\right]^{1/p} \left[{\sum_{n=1}^\infty (a_n)^q}\right]^{1/q}$$ (3)

and

$$\int_0^\infty \int_0^\infty {f(x)g(y)\over (x+y)^\lambda}\,d... ...\right)^{1/p} \left({\int_0^\infty [g(x)]^q\,dx}\right)^{1/q}.$$ (4)

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

See also Hilbert's Inequality

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.