Simultaneous Approximation by. Complex Polynomials

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 36, HIKARI Ltd, Simultaneous Approximation by Complex Polynomials Eman Samir Bhaya and Asmaa Raheem Hadi University of Babylon, College of Education for Pure Sciences, Iraq Copyright 2013 Eman Samir Bhaya and Asmaa Raheem Hadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we prove the following theorem for simultaneous approximation in L spaces for p<1:let m N. For any f A () and nm, there exists a polynomial P of degree, such that for all j=0,,mwehave f Pn cpn E f where cp0 depends on p butit isindependent of n and f. Here E f ) =inf{ f P, P is polynomial of degree n} Keywords: complex approximation, shape preserving approximation, interpolateing polynomials 1. Introduction As a fundamental definitions In our paper we have, = {z ; z <1}be the open unit disk and lete f inf fp,be the degree of best approximation of f by P, where Pis polynomial of degree n}.y best approximation to f from y if and only if fy inf fw,w Y.the following two results are known:

2 1804 Eman Samir Bhaya and Asmaa Raheem Hadi Theorem 1.1[6]: Let m.for any f A and nm, there exists a polynomial P of degree n, such that for all j = 0,..., m we have f z p z A ω f ;, z D, wherea is independent of nandz. Here ω g;δsup fufv ;u,v, uv δ. Theorem 1.2[1]: Let us suppose that m, q, r,f A and consider the distinct points z l = 1, l= 1,..., q. Then, for any n,nqmr,there exists a polynomial p of degreen, such that for all j = 0,..., mwe have f z p z c ω f ;, z,and p z f z,l1,,q,where C is independent of n and z.hereω g;δ:sup E g; Bz;δ,B(;δ ξ ; ξ δ,e g;m:inf gp ;P complex polynomial of degree m.theorem 1.1was proved by (N.N.Vorob ev) in [6] for the so-called domains of type1.1in the complex plane (including the unit disk) and Theorem1.2 was proved by (V.V. Andrievskii, I. Pritsker and R. Varga)in [1] for general continuain the complex plane (including the unit disk).in our paper, we will re-state them here for the particular case of unit disk only. Unfortunately, the constants appearing in these estimates, are claimed in the corresponding papers, as independent of n and zonly, without to be mentioned the independence of ftoo. Because of complicated technical details, it seems to be very difficult to deduce from the proofs of Theorems 1.1and 1.2that possibly these constants are independent of f too. For this reason, in the case of unit disk, by our main result Theorem 3.1, in the following section we improve these theorems to the spaces L,for p1 with new simple proof which clearly shows that the constant is independent of n,z and f too. 2. The Auxiliary results To prove our theorem we need the following definitions and results for f A,σ, fz T fz,where T fz z! Lemma 2.1: T fzt f z. Proof: T fz z,k0,,mt fz j z j z!!! z T f )(z) Lemma 2.2: σ, fzσ, f z!

3 Simultaneous approximation by complex polynomials 1805 Proof: σ, fz,k=0,,m,σ, fz bylemma2.1weget σ, fz T fzwheren2n,hnm T fz 1 nm1 σ 1,fz nm1 σ 1,fz nm1 σ, fz 1 nm1 T f zσ, f z T f zσ, f z T f zσ, f z T f zσ, f z Lemma 2.3 [5]: fσ, f A where:e f inf f p,p is polynomail of degree},a is an absolute constant independent of f, n and m. Lemma 2.4[3]: sup, p x p, Proposition 2.5: fσ, f cp,0 where cpis an absolute constant independent of f,n and m. Proof: It is sufficient to show E f cpe f Assume E f fq, 0, 2 n AndP be apolynomial of best approximation of f.then we may writefp p p.thus S.N.Bernstein inequality [2] yieldse f p Then by lemma 2.4we obtain p E f cp p p cp fq 1

4 1806 Eman Samir Bhaya and Asmaa Raheem Hadi cpe f Hence fσ, f fσ, f A Then from(1)we get f σ, f cp Lemma 2.6: Ifn1,then E f E f Proof: E f inf fp, where inf fp E f Lemma 2.7: fσ, f cpe f where cpis an absolute constant independent of f,n and m. Proof: UsingProposition 2.5 to obtain fσ, f cp E f nmj1 cpe f 1 nmj1 cpe f 2nk1 nm1 bynmsothat fσ, f cpe f cp2m 1E f Lemma 2.8[4]: E f cpn E f where cpis a constant depending only on p. Proposition 2.9: E f cpn E f where cpis a constant depending only on p. Proof: Using lemma2.8 we have f cpe f cpn E f Using the same lines used for the proof of proposition 2.5we can get E f cpe f cpn E f cpn E f 3. The main result Theorem 3.1: f P cpn E f

5 Simultaneous approximation by complex polynomials 1807 where cpis a constant depending only on p. Proof: f P f σ, f σ, fp Usinglemma2.2weget f P f σ, f σ, fp Usinglemma2.7weobtain f P cpe f σ, fp by Bernstein s inequality we have f P cpe f 2n σ, fp cpe f Cpn σ, ff f P cpe f Cpn E fe f cpe f Cpn E f byproposition2.9weget f P CpE f Cpn n E f by Proposition 2. 9we obtain,for nmk f P Cpnmk E f Cpn E f Cpn E f References [1] V.V.Andrievskii,I.PritskerandR.Varga,Simultaneous approximation and interpolation of functions on continua in the complex Plane, J. Math. Pures Appl.,80(2001), [2] E.S. Bhaya,On the constrained and unconstrained approximation, P. D. Thesis Baghdad University (2003). [3] R.A.Devore, D. Leviatan and X.M.Yu, Polynomial approximation in L 0 1,Constr. Approx., 8 (1992), [4] S.G.Gal,Shape Preserving approximation by complex polynomials in the unit disk, vol. 3(2008),No. 2, pp [5] S.B.Stechkin, On the approximation of Periodic functions by the de la Vall eepoussin sums, Anal. Math., 4 (1978),61-78.

6 1808 Eman Samir Bhaya and Asmaa Raheem Hadi [6] N.N.Vorob ev, Simultaneaous approximation of functions and their derivatives in a complex domain, Ukrainian Mathematical Journal, 20(1968),No ; (translated from UkrainskiiMatematicheskiiZhurnal, vol. 20 (1968, No. 1,pp ). Received: April 10, 2013