Title
On The Shape Preserving Approximation: Constrained and Unconstrained Approximation,Used
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Sometimes one may desire to approximate a function defined on a finite interval (for example [1,1]), subject to the conservation of so called shape properties (positivity, monotonicity and convexity). The first contribution is that we have approximated a function from a space Lp[1,1], 0 > p, by a number of piecewise linear functions and we have obtained global estimate of each of them using the second order of Ditzian Totik modulus of smoothness. Furthermore, these piecewise linear functions preserves the positivity of the function. Also proved the rate of coconvex approximation in the Lp[1,1] spaces, in terms of the third order of Ditzian Totik modulus of smoothness, where the constants involved depend on the location of the points of change of convexity. We have thus filled up a gap due to the uncertainty between previously known estimates involving the second order of Ditzian Totik modulus of smoothness and the impossibility of having such estimates involving with the second order of usual modulus of smoothness.
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