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Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. Interpolation processes [electronic resource] : basic theory and applications. Responsibility Giuseppe Mastroianni, Gradimir V. Imprint Berlin : Springer, c Physical description xiv, p.

Series Springer monographs in mathematics. Online Available online.

The Annals of Probability

More options. Find it at other libraries via WorldCat Limited preview. Bibliography Includes bibliographical references p. Contents 1. Constructive Elements and Approaches in Approximation Theory. Orthogonal Polynomials and Weighted Polynomial Approximation. Trigonometric Approximation. Algebraic Interpolation in Uniform Norm.

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There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P.

Vertesi: Interpolation of Functions, published in by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G. Lorentz, K. Jetter, S.

Interpolation process - Encyclopedia of Mathematics

We must be clear about terminology. We shall rarely use the term Chebyshev approximation , for that expression refers specifically to an approximation that is optimal in the minimax sense.

Convergence & Divergence - Geometric Series, Telescoping Series, Harmonic Series, Divergence Test

Chebyshev approximations are fascinating, and in Section 4. These approximations are not quite optimal, but they are nearly optimal and much easier to compute. Thus we can print the coefficients of the first few Chebyshev polynomials like this:. Note that that output of "poly" follows the pattern for Matlab's standard "poly" command: it is a row vector, and the high-order coefficients come first. So long as f is continuous and at least a little bit smooth Lipschitz continuity is enough , it has a unique expansion of this form, which converges absolutely and uniformly, and the coefficients are given by the integral.

One way to approximate a function is to form the polynomials obtained by truncating its Chebyshev expansion,. This isn't quite what Chebfun does, however, since it does not compute exact Chebyshev coefficients. The system actually stores a function by its values at the Chebyshev points rather than its Chebyshev coefficients, but this hardly matters to the user, and both representations are exploited for various purposes internally in the system.

Conversely, if f is a chebfun, then chebpoly f is the vector of its Chebyshev coefficients. Like "poly", "chebpoly" returns a row vector with the high-order coefficients first. If we apply chebpoly to a function that is not "really" a polynomial, we will usually get a vector whose first entry i. This reflects the adaptive nature of the Chebfun constructor, which always seeks to use a minimal number of points. By using "poly" we can print the coefficients of such a chebfun in the monomial basis.

Here for example are the coefficients of the Chebyshev interpolant of exp x compared with the Taylor series coefficients:. The fact that these differ is not an indication of an error in the Chebfun approximation. On the contrary, the Chebfun coefficients do a better job of approximating than the truncated Taylor series.

We can examine the approximation qualities of Chebyshev interpolants by means of a command of the form "chebfun When an integer N is specified in this manner, it indicates that a Chebyshev interpolant is to be constructed of precisely length N rather than by the usual adaptive process. Let us begin with a function that cannot be well approximated by polynomials, the step function sign x. To start with we interpolate it in 10 or 20 points, taking N to be even to avoid including a value 0 at the middle of the step. We can zoom in on the overshoot region by resetting the axes:.

The second plot is jagged, not because there is anything wrong with the underlying chebfun but because we have zoomed in very closely on the result of a "plot" command. Another way would be to use the 'interval' flag in the "plot" command. What is the amplitude of the Gibbs overshoot for Chebyshev interpolation of a step function? We can find out by using "max":.

What this means for Chebfun is that so long as a function is twice continuously differentiable, it can usually be approximated to machine precision for a workable value of N, even without subdivision of the interval. After the step function, a function with "one more derivative" of smoothness would be the absolute value. For example:. Such plots look good to the eye, but they do not achieve machine precision. We can confirm this by using "splitting on" to compute a true absolute value and then measuring some norms.

It is interesting to plot convergence as a function of N. According to Theorem 2 of the next section, this happens because f has a fifth derivative of bounded variation. Here is an example of a smoother function, one that is in fact analytic. According to Theorem 3 of the next section, if f is analytic, its Chebyshev interpolants converge geometrically. In this example we take f to be the Runge function, for which interpolants in equally spaced points would not converge at all in fact they diverge exponentially -- see Section 4.

Interpolation errors

This time the convergence is equally clean but quite different in nature. Now the straight line appears on the semilogy axes rather than the loglog axes, revealing the geometric convergence. The mathematics of Chebfun can be captured in five theorems about interpolants in Chebyshev points. If Let f, f', Smoother than this would be a C-infty function, i.

In such a case the convergence is geometric. The essence of the following theorem is due to Bernstein in , though I do not know where an explicit statement first appeared in print. The next theorem asserts that Chebyshev interpolants can be computed by the barycentric formula [Salzer ]. The final theorem asserts that the barycentric formula has no difficulty with rounding errors.

Our "theorem" is really just a placeholder; see [Higham ] for a precise statement and proof. Nevertheless it is provably stable. For practical computations, it is rarely worth the trouble to compute a best minimax approximation rather than simply a Chebyshev interpolant. Nevertheless best approximations are a beautiful and well-established idea, and it is certainly interesting to be able to compute them.

Chebfun makes this possible with the command "remez", named after Evgeny Remez, who devised the standard algorithm for computing these approximations in For example, here is a function on the interval [0,4] together with its best approximation by a polynomial of degree Note that a second output argument from remez returns the error as well as the polynomial. Let's add the error curve for the degree 20 i. Notice that although the best approximation has a smaller maximum error, it is a worse approximation for almost all x.

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Chebfun "remez" command can compute certain rational best approximants too, though it is somewhat fragile. CF approximation often comes close to optimal for non-smooth functions too, provided you specify a fourth argument to tell the system on how fine a Chebyshev grid to sample:.


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Chebfun is based on polynomial interpolants in Chebyshev points, not equispaced points.