ACTA issues

On convergent interpolatory processes associated with periodic-basis functions

F. J. Narcowich, N. Sivakumar, J. D. Ward

Acta Sci. Math. (Szeged) 68:1-2(2002), 133-161

Abstract. A periodic-basis function (PBF) is a function of the form $$ \phi(u)=\sum_{k\in{\msbm Z}}\widehat{\phi }(k) e^{iku}, u\in{\msbm R}, $$ where the sequence of Fourier coefficients $\{\widehat{\phi }(k) : k\in{\msbm Z}\} $ satisfies the following conditions: $$ \widehat{\phi }(k)=\widehat{\phi }(-k), k\in{\msbm Z}, \hbox{ and } \sum_{k\in{\msbm Z}}|\widehat{\phi }(k)|< \infty. $$ A PBF $\phi $ is said to be strictly positive definite if every Fourier coefficient of $\phi $ is positive. It is known that if $\phi $ is strictly positive definite, then given any continuous $24$-periodic function $f$ and any triangular array $\{\theta_{j,\mu } : 1\le j\le\mu, \mu\in {\msbm N}\} $ of distinct points in $[-4,4)$, there exists a unique PBF interpolant $ I(\theta ):= \sum_{j=1}^\mu a_j\phi(\theta -\theta_{j,\mu })$, $a_j\in{\msbm R}$, such that $ I(\theta_{k,\mu })=f(\theta_{k,\mu })$, $1\le k\le\mu $. This paper studies the uniform convergence of these PBF interpolants to the approximand $f$. Even though there is a rather well-developed theory which supplies various results of this nature, it also has the shortcoming that if $\phi $ is very smooth, then the class of functions $f$ which can be simultaneously approximated and interpolated by PBF interpolants is highly restricted. The primary objective of this paper is to suggest an oversampling strategy to overcome this problem. Specifically, it is shown that by increasing the dimension of the underlying space of approximants/interpolants judiciously, one can construct PBF interpolants (based on very smooth $\phi $) that converge to approximands which are only assumed to be continuous. The main tool in the analysis is a periodic version of a result of Szabados on algebraic polynomials, the proof of which relies on the trigonometric version of a fundamental theorem due to Erdős.

AMS Subject Classification (1991): 41A05, 41A30, 42A08, 42A12

Received November 21, 2000, and in revised form July 17, 2001. (Registered under 2834/2009.)