Interpolation by periodic radial basis functions

• Published in: Journal of mathematical analysis and applications Vol. 168; no. 1; pp. 111 - 130
• Main Authors: Light, W.A, Cheney, E.W
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.07.1992; Elsevier
• Subjects: Approximations and expansions; Exact sciences and technology; Mathematical analysis; Mathematics; Sciences and techniques of general use; Interpolation; Periodic function; Necessary and sufficient condition; Rachial basis function
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/0022-247X(92)90193-H

On the unit circle S 1, let d be the natural (geodesic) metric. We investigate the possibility of interpolating arbitrary data on a set of nodes y i ϵ S 1 by means of a function of the form x ↦ ∑ i = 1 n c i f( d( x, y i )). Here f is a function from [0, π] to R , and is subject to our choice. The interpolation matrix A having elements A ij = f( d( y j , y i )) is crucial to this problem. In the basic case, f( x) = x, we give necessary and sufficient conditions on the nodes for the invertibility of A. For equally-spaced nodes, we give nearly complete conditions on f for the invertibility of A.