Rational approximation of analytic functions

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It only takes a minute to sign up. I know that every continuous function can be approximated by sequence of polynomials and given function is analytic in annulus A. But how to construct such sequence of functions? Consider the Runge's Theorem and its corollary about polynomials. Sign up to join this community. The best answers are voted up and rise to the top.


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Home Questions Tags Users Unanswered. Sequence of functions which approximate analytic function Ask Question. Asked 2 years, 3 months ago. Active 2 years, 3 months ago. Viewed times. Which of the following are true? Can we use identity theorem?

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Robert Z k 10 10 gold badges 79 79 silver badges bronze badges. Unfortunately the polynomials Received 3 February The research of the first author was supported, in part, by the National Science Foundation, under grant DMS The research of the second author was partially supported by the Hungarian National Science Foundation for Research, grant no. London Math. This shortcoming lends support to the principle of contamination in best polynomial approximation, which was introduced by the first author in [9]. In fact, for our construction, the inequality 1.

As an application of our results we shall prove a conjecture of R. Grothmann and E. The outline of the paper is as follows. In Section 2, we discuss the best possible rate of polynomial approximation to the sign function. In Section 3, we use the approximations to the sign function to construct a sequence of polynomials pn having properties i and ii above. We also show that the rate of convergence obtained is best possible.

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Finally, in Section 4, we consider polynomial approximants in regions of the complex plane to the function g z given in 1. Polynomial approximation of the sign function The following question arises in connection with several problems of approxi- mation theory cf. More precisely, we shall prove the following. Assuming, for the moment, that Theorem 2 is true, we shall give the following. Proof of Theorem 1. Suppose first that 2.

Then 2. Suppose now that 2. We assume, as we may, that the polynomials Qn satisfying 2.

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From 2. Since Pn x is odd, inequality 2.

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Proof of Theorem 2. We first establish the necessity of 2. But, by the well-known inequality of S.

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Bernstein cf. We now establish the sufficiency of 2. This construction is well known in the theory of orthogonal polynomials associated with Freud exponential weights cf. Hence our proof is complete. Approximation of piecewise analytic functions Our goal is to construct a sequence of polynomial approximants having the properties i to iii described in the introduction. We shall prove the following. Integrating 3. With this choice we get from 3. We have actually proved the following more general result.

Let a,ft satisfy condition 2. Writing it follows from 3. Bernstein see, for example, [1, Appendix 42, p.

Numerical Solution for the Extrapolation Problem of Analytic Functions

From 3. Thus, g is an entire function of exponential type 1 which is bounded on the real line; hence [1, Chapter 4] z , zeC. This and 3.