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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.