رسائل دكتوراة
English
2022
Pointwise Polynomial Approximation
إيمان سمير عبد علي بهية, اكتفاء ضياء جليل كريم
كلية التربية للعلوم الصرفة-جامعة بابل · العراق
In our dissertation we study the shape preserving for convex and piecewise convex functions in quasi normed spaces LpL_pLp for 0<p<10 < p < 10<p<1.
Many authors studied the approximation of convex continuous functions by algebraic polynomials. We generalize and improve their works for piecewise algebraic polynomials and study the approximation in LpL_pLp space for 0<p<10 < p < 10<p<1, we prove for a given convex function in LpL_pLp, 0<p<10 < p < 10<p<1, there exists a convex piecewise polynomial as a convex approximation of F\mathcal{F}F. Then we introduce some approximating properties for these approximations.
Some papers introduced the direct and inverse theorems for the piecewise convex continuous functions. We study the piecewise convex approximation of function in LpL_pLp, 0<p<10 < p < 10<p<1 using piecewise polynomials. It means if F\mathcal{F}F changes its convexity at {yj}j=1r\{y_j\}_{j=1}^{r}{yj}j=1r, we can construct a piecewise convex piecewise polynomial as a best approximation change its convexity at {yj}j=1r\{y_j\}_{j=1}^{r}{yj}j=1r. Our degree of approximation is in terms of the ordinary modulus of smoothness.
We introduce Jackson type theorem for piecewise convex functions in LpL_pLp quasi normed space LpL_pLp with 0<p<10 < p < 10<p<1. Using piecewise convex piecewise polynomials in terms of Ditzian Totik modulus of smoothness.
