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عنوان البحث(Papers / Research Title)

Interpolating Operators for Multiapproximation

الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)

ايمان سمير عبد علي بهية

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ايمان,سمير,عبد,علي,بهية ,Interpolating Operators for Multiapproximation , Time 5/17/2011 5:56:14 PM : كلية التربية للعلوم الصرفة

وصف الابستركت (Abstract)

Interpolating Operators for Multiapproximation

الوصف الكامل (Full Abstract)

Interpolating Operators for Multiapproximation Eman Samir Bhaya Department of Mathematics, College of Education-Ibn Hayyan, Babylon University, Iraq   Abstract:     Problem Statement: There are no simple definitions of operators for best multiapproximation and best one sided multiapproximation which work for any measurable function in Lp for, p>0 .    This paper investigates operators that are good for best multiapproximation and best one sided multiapproximation. Approach: We first introduce some direct results related to the approximation problem of continuous functions by Hermit-Fejer interpolation based on the zeros of Chebyshev polynomials of the first or second kind in terms of the usual modulus of continuity.   They are then improved to spaces Lp for p<1, in terms of the first order averaged modulus of smoothness. However, because this suffers from convergence problems, we improve and generalize these direct estimations by defining an interpolating multivariate operator Hn(f) of measurable functions, that operator based on the zeros of Chepyshev polynomials of the first kind and prove that for any measurable  function defined on Lp[-1,1 ]d  the sequence Hn(f) converges uniformly to f.   Results and Discussion: The resulting operators were defined for functions f such that f(k), k = 0,1,…  is of bounded variation. Then, the order of best onesided trigonometric approximation to bounded measurable functions in terms of the average modulus of smoothness was characterized. Estimates characterizing the order of best onesided approximation in terms of the k-th averaged modulus of smoothness for any function in spaces Lp, p<? were obtained.   In our work we also approximate onesidedly these measurable functions in Lp[-1,1]d by defining a new operator for onesided approximation and prove a direct theorem for best onesided multiapproximation in terms of the first order averaged moduli of smoothness. Conclusion: The proposed  method successfully construct operators for best multi approximation and best onesided multiapproximation for any measurable function in Lp for , p>0. Key words:    Interpolating operator, best approximation, onesided approximation, multiapproximation

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