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A note on the weighted Hilbert's inequality
Author(s):
Xian-Jin
Li
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1165-1173.
MSC (2000):
Primary 47B32, 46E22
Posted:
October 14, 2004
MathSciNet review:
2117219
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Abstract:
A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert's inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.
References:
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- E. Hellinger and O. Toeplitz, Grundlagen für eine Theorie der unendlichen Matrizen, Math. Ann. 69 (1910), 289-330.
- 3.
- Xian-Jin Li, On reproducing kernel Hilbert spaces of polynomials, Math. Nachr. 185 (1997), 115-148.MR 98g:46031
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- Xian-Jin Li, An explicit formula for finite Hilbert transforms associated with a polynomial, Indiana Univ. Math. J. 53 (2004), 185-203. MR 2048189
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- H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. 8 (1974), 73-82. MR 49:2544
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Additional Information:
Xian-Jin
Li
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
xianjin@math.byu.edu
DOI:
10.1090/S0002-9939-04-07606-3
PII:
S 0002-9939(04)07606-3
Keywords:
Hilbert transforms,
reproducing kernel Hilbert spaces
Received by editor(s):
October 21, 2003
Received by editor(s) in revised form:
December 3, 2003
Posted:
October 14, 2004
Additional Notes:
This research was supported by National Security Agency MDA 904-03-1-0025
Communicated by:
Joseph A. Ball
Copyright of article:
Copyright
2004,
American Mathematical Society
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