Discrete Linear Interpolatory Operators

This is a survey on discrete linear operators which, besides approximating in Jackson or near-best order, possess some interpolatory property at some nodes. Such operators can be useful in numerical analysis. A central problem in approximation theory is the construction of simple functions that approximate well a given set of functions. Traditionally, by “simple” we mean polynomials or rational functions as they are easily implemented on computers, and the set of functions is generally characterized by continuity or belonging to some L class. These functions may be defined on finite or infinite intervals, or in complex domains. Another characteristic is the measure of error which can be the supremum or L norm, etc., with a weight usually introduced in the case of non-compact domains. But what are the available data to construct such approximations? From a practical point of view, this is a crucial question. For example, convolution integrals are very useful in proving Jackson’s theorem, but to actually calculate them one needs complete knowledge of the function. Obviously, this precludes any numerical application. In practice, we are generally given a discrete set of data (like function values at certain points) from which we wish to reconstruct the function. Another feature which is important to us is that we prefer to construct linear operators since they are easier to handle. For example, the so-called “best approximation” is not at all “best” for our purposes, since (except in inner product spaces) it represents a non-linear operator which may be very difficult to calculate. A natural candidate that satisfies the above requirements is some kind of interpolation operator. It is based on a discrete set of data, and additionally gives a zero error at an increasing number of well defined points. Depending on the nature of the interpolation, it is relatively easy to construct such interpolation operators. However, there is a significant drawback to these operators. In the case of Lagrange interpolation, for example, no matter how we choose the n nodes we get at least an extra O(log n) factor compared to the optimal Jackson order of convergence. And if we try to avoid this problematic situation by considering Hermite–Fejér interpolation then, although we obtain the Jackson order of convergence in some cases, this process is saturated. That is, it will not give the Jackson order of approximation, for example, for Lip 1 classes, let alone for classes of functions with higher order of smoothness. Surveys in Approximation Theory 53 Volume 2, 2006. pp. 53–60. Copyright oc 2006 Surveys in Approximation Theory. ISSN 1555-578X All rights of reproduction in any form reserved. J. Szabados 54 It transpires that the reason for this negative behavior of interpolating polynomials is the strict restriction on the degree. In 1963, at an Oberwolfach conference, Paul Butzer raised the following question: Is it possible to construct an interpolation process that gives the Jackson order of convergence? Of course, this was meant in the sense that we should now allow interpolating polynomials of degree higher than Lagrange or Hermite–Fejér interpolation, say polynomials of degree ≤ an for some constant a. It was Géza Freud [11] who answered this question affirmatively by constructing polynomials pn of degree at most 4n − 3, interpolating at ∼ n/3 nodes and approximating to the Jackson order max |x|≤1 |f(x)− pn(x)| ≤ c ω(f, 1/n), |x| ≤ 1, where ω is the ordinary modulus of continuity. This was later improved upon by Freud and Vértesi [14] who constructed a sequence of discrete linear polynomial operators (DLPO, in what follows) Jn(f, x) of degree at most 4n − 2 which interpolate at the Chebyshev nodes cos((2k − 1)/(2n))π, k = 1, . . . , n, and provide a Timan type (pointwise) estimate |f(x)− Jn(f, x)| ≤ c[ω(f, √ 1− x2 n ) + ω(f, 1 n )], |x| ≤ 1. (1) Freud and Sharma [12, 13] further extended this result to operators based on general Jacobi nodes, and also succeeded in decreasing the degree of the polynomial to n(1 + ε), for an arbitrary ε > 0. (Of course, the c in the above then depends upon ε.) Freud’s work [11] initiated a substantial series of papers exhibiting constructions of a similar character. R. B. Saxena [19] succeeded in constructing a sequence of DLPOs J n which realized the even stronger Telyakovskii–Gopengauz estimate |f(x)− J n(f, x)| ≤ c ω(f, √ 1− x2 n ), |x| ≤ 1. Furthermore, for 2π-periodic continuous functions, O. Kis and P. Vértesi [17] constructed a very simple interpolating process: Let lk(x) be the fundamental functions of trigonometric interpolation based on the equidistant nodes xk = 2kπ/(2n+1), k = 0, . . . , 2n, i.e., let lk(x) be that trigonometric polynomial of order n for which lk(xj) = δjk, j, k = 0, . . . , 2n, and consider the sequence of operators Un(f, x) = n