Degree of best one-sided approximation by smooth multi-Dimensional spline in weighted spaces

Abstract

Degree of best one-sided approximation of multivariate function f that lies in weighted space (Lp,α − space) by construct an operator S∓ n which is dependent on constructed multi-dimensional spline s satisfies: ‖ f − s‖∞,α(Ω) ≤ c ∑ ∝∈∂Λ δ ∝ ‖D∝ f‖∞,α(Ω) Where the domain is coordinate wise convex(Ω = [−1,1]) has been studied in this work.The result which we end in it that the degree of best one-sided approximation of the function f which lies in the Sobolev space W Λ p (Ω)(i.e. f and D∝ f lies in Lp,α ) is less than when the function f lies in Lp,α . Also we estimate the degree of best one sided approximation of the function f ∈W Λ p (Ω) by multivariate modulus of derivatives of f by constrict intertwining pair of pairwise polynomials with respect to any partition xn which is called co-one sided approximation, the proof of this important result is depending on the same result when the function f in single case (i.e. f : R→ R) and properties of Cartesian product and projection of the real valued functions, also we use properties of modulus of smoothness in multivariate case. Finally we shall prove invers theorem of best one sided approximation of the function f by using an operator S∓ n (x) and the same result of best approximation of the function f and the relation between best and best one sided approximation.