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On moduli of smoothness of fractional order. (English) Zbl 1108.26004

The author investigates some properties of moduli of smoothness of order \(\beta\) (\(\beta>0\)) of a measurable \(2\pi\)-periodic function \(f\in L^p\) (\(1\leq p\leq +\infty\)) (continuous if \(p=+\infty\)). The modulus of smoothness \(\omega_{\beta}\) of \(f\) is defined via the differences of fractional order \(\beta\) as follows: \[ \omega_{\beta}(f,t)_p=\sup_{| h| \leq t}| | \triangle_h^{\beta}f(\cdot)| | _p, \] where \[ \triangle_h^{\beta}f(x)=\sum_{\nu=0}^{\infty}(-1)^{\nu}\binom{\beta}{\nu}f(x+(\beta-\nu)h). \] In the main result the author proves the equivalence of the modulus of smoothness \(\omega_{\beta}\) and a function in the class \(\Phi_{\beta},\) whose elements \(\varphi\) are nonnegative, bounded functions on \((0,+\infty)\) such that \(\varphi(t)\to 0\) as \(t\to 0,\) \(\varphi\) is nondecreasing and \(\varphi(t)t^{-\beta}\) is nonincreasing.
Reviewer: Rita Pini (Milano)

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
41A25 Rate of convergence, degree of approximation
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