Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article

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
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

1.: L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, NJ, 1968. MR 37:4590
2.: 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
4.: Xian-Jin Li, An explicit formula for finite Hilbert transforms associated with a polynomial, Indiana Univ. Math. J. 53 (2004), 185-203. MR 2048189
5.: H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. 8 (1974), 73-82. MR 49:2544
6.: I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1-28.
7.: J. D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985), 183-216. MR 86g:42005

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B32, 46E22

Retrieve articles in all Journals with MSC (2000): 47B32, 46E22

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


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS