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Hilbert's Constants

Extend Hilbert's inequality by letting p,q>1 and

1/p+1/q>=1,
(1)

so that

0<lambda=2-1/p-1/q<=1.
(2)

Levin (1937) and Stečkin (1949) showed that

sum_(m=1)^inftysum_(n=1)^infty(a_mb_n)/((m+n)^lambda)<={picsc[(pi(q-1))/(lambdaq)]}^lambda[sum_(m=1)^infty(a_m)^p]^(1/p)[sum_(n=1)^infty(a_n)^q]^(1/q)
(3)

and

int_0^inftyint_0^infty(f(x)g(y))/((x+y)^lambda)dxdy<{picsc[(pi(q-1))/p]}^lambda ×(int_0^infty[f(x)]^pdx)^(1/p)(int_0^infty[g(x)]^qdx)^(1/q).
(4)

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

See also

Hilbert's Inequality

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References

Finch, S. R. "Hilbert's Constants." §3.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 216-217, 2003.Levin, V. I. "On the Two-Parameter Extension and Analogue of Hilbert's Inequality." J. London Math. Soc. 11, 119-124, 1936.Levin, V. I. "Two Remarks on Hilbert's Double Series Theorem." J. Indian Math. Soc. 11, 111-115, 1937.Levin, V. I. "Two Remarks on Van Der Corput's Generalisations of Knopp's Inequality." Kon. Akad. van Wetensch. Amsterdam 40, 429-431, 1937.Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, 1991.Stečkin, S. B. "On the Degree of Best Approximation to Continuous Functions." Dokl. Akad. Nauk SSSR 65, 135-137, 1949.

Referenced on Wolfram|Alpha

Hilbert's Constants

Cite this as:

Weisstein, Eric W. "Hilbert's Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertsConstants.html

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