convergence of the reconstruction formula for the potential function

Authors:
Ya-Ting Chen, Y. H. Cheng, C. K. Law and J. Tsay

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2319-2324

MSC (2000):
Primary 34A55; Secondary 34B24

DOI:
https://doi.org/10.1090/S0002-9939-02-06297-4

Published electronically:
January 17, 2002

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Abstract | References | Similar Articles | Additional Information

Abstract: It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact .

**1.**O.H. Hald and J.R. McLaughlin,*Solutions of inverse nodal problems*, Inverse Problems,**5**(1989), 307-347. MR**90c:34015****2.**O.H. Hald and J.R. McLaughlin,*Inverse problems: finding the potential from nodal lines*, AMS Memoir,**119**, No. 572 (1996). MR**97d:35240****3.**O.H. Hald and J.R. McLaughlin,*Inverse problems: recovery of BV coefficients from nodes*, Inverse Problems,**14**(1998), 245-273. MR**99b:34028****4.**C.K. Law, C.L. Shen and C.F. Yang,*The inverse nodal problem on the smoothness of the potential function*, Inverse Problems,**15**(1999), 253-263. MR**2000a:34020****5.**C.K. Law, C.L. Shen and C.F. Yang,*Errata: The inverse nodal problem on the smoothness of the potential function*, Inverse Problems,**17**(2001), no. 2, 361-363.**6.**C.K. Law and J. Tsay,*On the well-posedness of the inverse nodal problem*, Inverse Problems,**17**(2001), no. 5, 1493-1512.**7.**C.J. Lee and J.R. McLaughlin,*Finding the density for a membrane from nodal lines*, Inverse Problems in Wave Propagation, eds. G. Chavent et al., Springer, New York (1997), 325-345.MR**98h:35236****8.**J.R. McLaughlin,*Inverse spectral theory using nodal points as data - a uniqueness result*, J. Diff. Eqns.**73**(1988), 354-362.MR**89f:34035****9.**J.R. McLaughlin,*Solving inverse problems with spectral data*, Surveys on Solution Methods for Inverse Problems, eds. D. Colton et al., Springer, New York (2000), 169-194. MR**2001f:35431****10.**C.L. Shen,*On the nodal sets of the eigenfunctions of the string equations*, SIAM J. Math. Anal.,**19**(1988), 1419-1424.MR**89j:34035****11.**C.L. Shen,*On the nodal sets of the eigenfunctions of certain homogeneous and nonhomogeneous membranes*, SIAM J. Math. Anal.,**24**(1993) 1277-1282.MR**94i:35138****12.**C.L. Shen and T.M. Tsai,*On uniform approximation of the density function of a string equation using eigenvalues and nodal points and some related inverse problems*, Inverse Problems,**11**(1995), 1113-1123.MR**96g:34021****13.**X.F. Yang,*A solution of the inverse nodal problem*, Inverse Problems,**13**(1997), 203-213.MR**98c:34017**

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Additional Information

**Ya-Ting Chen**

Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China

Email:
chenyt@math.nsysu.edu.tw

**Y. H. Cheng**

Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China

Email:
jengyh@math.nsysu.edu.tw

**C. K. Law**

Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China

Email:
law@math.nsysu.edu.tw

**J. Tsay**

Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China

Email:
tsay@math.nsysu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9939-02-06297-4

Keywords:
Inverse nodal problem,
potential function,
reconstruction formula

Received by editor(s):
November 21, 2000

Received by editor(s) in revised form:
February 20, 2001

Published electronically:
January 17, 2002

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society