Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A new interpolation formula for the Titchmarsh-Weyl $ m$-function


Authors: Alexei Rybkin and Vu Kim Tuan
Journal: Proc. Amer. Math. Soc. 137 (2009), 4177-4185
MSC (2000): Primary 47E05, 65D05
DOI: https://doi.org/10.1090/S0002-9939-09-09983-3
Published electronically: June 25, 2009
MathSciNet review: 2538578
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the Titchmarsh-Weyl $ m$-function of the half-line Schrödinger operator with Dirichlet boundary conditions we put forward a new interpolation formula which allows one to reconstruct the $ m$-function from its values on a certain infinite set of points for a broad class of potentials.


References [Enhancements On Off] (What's this?)

  • 1. J. Agler and J. E. McCarthy. Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, 44. American Mathematical Society, Providence, RI, 2002. MR 1882259 (2003b:47001)
  • 2. S. Avdonin, V. Mikhaylov, and A. Rybkin. The boundary control approach to the Titchmarsh-Weyl $ m$-function. I. The response operator and the $ A$-amplitude. Comm. Math. Phys. 275 (2007), no. 3, pp. 791-803. MR 2336364 (2008g:93083)
  • 3. A. Boumenir and Vu Kim Tuan. The interpolation of the Titchmarsh-Weyl function. J. Math. Anal. Appl. 335 (2007), pp. 72-78. MR 2340306 (2008f:34059)
  • 4. A. Boumenir and Vu Kim Tuan. Sampling eigenvalues in Hardy spaces. SIAM J. Numer. Anal. 45 (2007), no. 2, pp. 473-483. MR 2300282 (2008c:65180)
  • 5. F. Gesztesy and B. Simon. A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure. Ann. of Math. (2) 152 (2000), no. 2, pp. 593-643. MR 1804532 (2001m:34185b)
  • 6. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev. Integrals and Series. Special Functions (Russian). Nauka, Moscow, 1983, English translation: Integrals and series. Vol. 3. More special functions. Gordon and Breach Science Publishers, New York, 1990. MR 1054647 (91c:33001)
  • 7. A. G. Ramm. Recovery of the potential from $ I$-function. C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 4, pp. 177-182. MR 896977 (88k:34012)
  • 8. B. Simon. A new approach to inverse spectral theory, I. Fundamental formalism. Ann. of Math. (2) 150 (1999), no. 3, pp. 1029-1057. MR 1740987 (2001m:34185a)
  • 9. P. K. Suetin. Classical Orthogonal Polynomials (Russian), Second edition. Nauka, Moscow, 1979. MR 548727 (80h:33001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47E05, 65D05

Retrieve articles in all journals with MSC (2000): 47E05, 65D05


Additional Information

Alexei Rybkin
Affiliation: Department of Mathematics and Statistics, University of Alaska Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775
Email: ffavr@uaf.edu

Vu Kim Tuan
Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118
Email: vu@westga.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09983-3
Keywords: Titchmarsh-Weyl $m$-function, interpolation, $A$-amplitude, Hardy space
Received by editor(s): October 28, 2008
Received by editor(s) in revised form: April 2, 2009
Published electronically: June 25, 2009
Additional Notes: This research was supported in part by the U.S. National Science Foundation under grant DMS 070747
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society