We generalize the Zygmund inequality for the conjugate function to the Morrey type spaces defined on the unit circle T. We obtain this extended Zygmund inequality by introducing the Morrey-Zygmund space on T.
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For 2π-periodic functions from Lp (where 1 < p < ∞) we prove an estimate of approximation by Euler means in Lp metric generalizing a result of L. Rempuska and K. Tomczak. Furthermore, we show that this estimate is sharp in a certain sense. We study the uniform approximation of functions by Euler means in terms of their best approximations in p-variational metric and also prove the sharpness of this estimate under some conditions. Similar problems are treated for conjugate functions.
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We extend and generalize the results of the first author [4]. Considering additionally conjugate functions and introducing a new subclass of integrable functions we obtain the results of the L. Leindler [3] and P. Chandra [1, 2] type.
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A conjugacy is introduced for the class of starshaped functions from [0, infinity] into [0, infinity], i.e. the class of functions f such that their slope s : t --> f (t)/t is nondecreasing. This class is stable by several operations and plays a key role in the study of uniformly convex and uniformly smooth convex functions and in the geometry of Banach spaces. Here the inversion of the subdifferential as in the Legendre-Fenchel transform is replaced by an inversion device of the slope s which uses the ordering of R.
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In this note we shall prove that if p is a power of 2, i.e. p = 2[sup k], k [belongs to] N, and f(0) = 0, that is f has the Fourier series Sigma[...] with a[sub o] = 0, then [...].
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