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1

Zygmund inequality of the conjugate function on Morrey-Zygmund spaces

Yee Tat-Leung, Ho Kwok-Pun

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

97--104

EN

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.

2

Approximation of functions and their conjugates in Lp and uniform metric by Euler means

Volosivets S. S., Tyuleneva A. A.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

141--150

EN

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.

3

On pointwise approximation of functions by some matrix means of conjugate Fourier series

Łenski W., Szal B.

Fasciculi Mathematici

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2015

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Nr 55

91--108

EN

The results corresponding to some theorems of S. Lal [Tamkang J. Math., 31(4)(2000), 279-288] and the results of the authors [Banach Center Publ. 92(2011), 237-247] are shown. The same degrees of pointwise approximation as in mentioned papers by significantly weaker assumptions on considered functions are obtained. From presented pointwise results the estimation on norm approximation with essentialy better degrees are derived. Some special cases as corollaries for iteration of the Norlund or the Riesz method with the Euler one are also formulated.

4

On the approximation of conjugate functions from Lp by some special matrix means of conjugate Fourier series

Krantz R., Rzepka A.

Commentationes Mathematicae

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2014

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Vol 54, No. 2

191--202

EN

The results corresponding to some theorems of W. Łenski and B. Szal are shown. From the presented pointwise results the estimates on norm approximation are derived. Some special cases as corollaries are also formulated.

5

On the degree of strong approximation of functions and conjugate functions

Łenski W., Szal B.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

117-128

EN

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.

6

A duality for starshaped functions

Penot J.-P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 2

127-139

EN

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.

7

A note on the M. Riesz theorem for the conjugate function

Papadopoulos S.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 3

283-288

EN

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 [...].