Approximation Theory: Moduli of Continuity and Global Smoothness Preservation
This monograph, in two parts, is an intensive and comprehensive study of the computational aspects of the moduli of smoothness and the Global Smoothness Preservation Property (GSPP). Key features include: * systematic and extensive study of the computation of Moduli of Continuity and GSPP, presented for the first time in book form * substantial motivation and examples for key results * extensive applications of moduli of smoothness and GSPP concepts to approximation theory, probability theory, numerical and functional analysis * GSPP methods to benefit engineers in computer-aided geometric design * good bibliography and index For researchers and graduate students in pure and applied mathematics.
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Global Smoothness Preservation
General Theory of Global Smoothness Preservation
Calculus of the Moduli of Smoothness
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2fcx algebraic polynomial analogous applications arccos assumptions Bernstein operators Bernstein polynomials bounded linear operator C(Rd Chapter classes of functions Cn[a conclusion consider constant continuous functions continuous probabilistic distribution convergence convex Corollary Definition En(f estimate example fixed fp(x Fubini's theorem functions from Rd Furthermore given global minimum point global smoothness preservation Gonska Hence i(arccosu immediately obtain inequality J—oc K-functional Lebesgue Lebesgue measurable Lemma Let f Let us assume let us define Let us denote Lorentz 194 mean value theorem metric space modulus of continuity modulus of smoothness monotone nonconcave nonconvex nondecreasing nonincreasing obviously Open problem polynomial of degree Popoviciu positive linear operators probabilistic distribution function proof of Theorem proves the theorem r+oc Remark respect Section sequence stochastic sup inf trigonometric polynomial u)du uniform norm uniformly continuous functions