×

On the Lagrange interpolation polynomials of entire functions. (English) Zbl 0535.41003

We investigate the growth of an entire function f and estimate the error term when approximating f in the complex plane by Lagrange interpolation polynomials over zeros of orthogonal polynomials. In particular, Lagrange interpolation at the zeros of Hermite polynomials is considered. Our main result is:
Theorem. Let W be the class of all weight functions of the form \(w_ Q(x)=\exp \{-2Q(x)\},x\in {\mathbb{R}}\), where (i) Q(x) is an even differentiable function, increasing for \(x>0\); (ii) there exists \(\rho<1\) such that \(x^{\rho}Q'(x)\) is increasing; and (iii) the unique positive sequence \(\{q_ n\}\) determined by \(q_ nQ'(q_ n)=n\) satisfies \(q_{2n}/q_ n=C>1\) for \(n=1,2,3,..\). for some constant C independent of n. Let f be an entire function, and \(M(R)=\max_{| z| =R}| f(z)|\). Let \(w_ Q\in W\). Then, there exists a constant \(A\in(0,1)\), depending on Q only, such that whenever \(\lim \sup_{R\to \infty}(\log \quad M(R)/2Q(R))\leq A\) we have for any \(z\in {\mathbb{C}}\), \(\lim \sup_{n\to \infty}| f(z)-{\mathcal L}_ n(w_ Q;f;z)|^{1/n}<1,\) where \(L_ n(w_ Q;f;z)=\sum^{n}_{k=1}f(z\chi_{kn})\ell_{kn}(z)\), \(\chi_{kn}\) are the zeros of the nth orthogonal polynomial associated with \(w_ Q\), and \(\ell_{kn}(z)\) are the fundamental polynomials of Lagrange interpolation.

MSC:

41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Jarrah, R., An error estimate for Gauss-Jacobi quadrature formula with the Hermite weight \(w(x) = exp\)(−\(x^2)\), Publ. Inst. Math. (Beograd), 33, 47, 17-22 (1983) · Zbl 0517.41023
[2] Boas, R. P., Entire Functions (1954), Academic Press: Academic Press New York
[3] Freud, G., Orthogonal Polynomials (1971), Pergamon: Pergamon Elmsford · Zbl 0226.33014
[4] Freud, G., On the greatest zero of orthogonal polynomial, I, Acta. Sci. Math., 34, 91-97 (1973) · Zbl 0262.33014
[5] Freud, G., On the greatest zero of orthogonal polynomial, II, Acta. Sci. Math., 36, 49-54 (1974) · Zbl 0285.33012
[6] Szegö, G., Orthogonal Polynomials (1959), Amer. Math. Soc: Amer. Math. Soc Providence, R.I · JFM 65.0278.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.