Hilbert,s Constants
المؤلف:
Finch, S. R.
المصدر:
"Hilbert,s Constants." §3.4 in Mathematical Constants. Cambridge, England: Cambridge University Press
الجزء والصفحة:
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26-2-2020
1372
Hilbert's Constants
Extend Hilbert's inequality by letting 
and
 |
(1)
|
so that
 |
(2)
|
Levin (1937) and Stečkin (1949) showed that
![sum_(m=1)^inftysum_(n=1)^infty(a_mb_n)/((m+n)^lambda)<=<span style=]() {picsc[(pi(q-1))/(lambdaq)]}^lambda[sum_(m=1)^infty(a_m)^p]^(1/p)[sum_(n=1)^infty(a_n)^q]^(1/q) " src="http://mathworld.wolfram.com/images/equations/HilbertsConstants/NumberedEquation3.gif" style="height:47px; width:376px" /> |
(3)
|
and
![int_0^inftyint_0^infty(f(x)g(y))/((x+y)^lambda)dxdy<<span style=]() {picsc[(pi(q-1))/p]}^lambda
×(int_0^infty[f(x)]^pdx)^(1/p)(int_0^infty[g(x)]^qdx)^(1/q). " src="http://mathworld.wolfram.com/images/equations/HilbertsConstants/NumberedEquation4.gif" style="height:82px; width:281px" /> |
(4)
|
Mitrinovic et al. (1991) indicate that this constant is the best possible.
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.
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