Effect of Double Weights on the Constrained Approximation in the Space Lp, p<1

SUMMARY
The book “Effect of double weights on the constraned approximation in the space Lp, p<1” we study double weighted co-positive approximation for functions in Lp,w quasi norm spaces. Many results introduced unconstrained approximation using the k-th order modulus of smoothness. Also very little were introduced in the double weighted spaces. Sometimes we need that the approximation has the same properties of the target function such as positivity, monotonicity, convexity and k- monotonicity. These constrains restrict very much the degree of the best approximation. We introduce direct theorem for co-positive approximation in terms of double weighted modulo of smoothness and we proved that by a negative theorem that the positivity restricts the order of approximation in terms of only. To improve the order of double weighted co-positive approximation we introduce direct theorem and equivalence for the un constrain approximation. We proved that there are certain conditions. We can achieve