On the Exponential Integrability of Conjugate Functions

H. Gissy; S. Miihkinen; J. A. Virtanen

doi:10.1007/s00041-021-09885-4

On the Exponential Integrability of Conjugate Functions

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09885-4 ◽

2021 ◽

Vol 27 (6) ◽

Author(s):

H. Gissy ◽

S. Miihkinen ◽

J. A. Virtanen

Keyword(s):

Conjugate Function ◽

Conjugate Functions ◽

Exponential Integrability ◽

Related Theorem

AbstractWe relate the exponential integrability of the conjugate function $${\tilde{f}}$$ f ~ to the size of the gap in the essential range of f. Our main result complements a related theorem of Zygmund.

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Hermite Conjugate Functions and Rearrangement Invariant Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-059-5 ◽

1973 ◽

Vol 16 (3) ◽

pp. 377-380 ◽

Cited By ~ 1

Author(s):

Kenneth F. Andersen

Keyword(s):

Weak Type ◽

Conjugate Function ◽

Poisson Integral ◽

Conjugate Functions ◽

Rearrangement Invariant Spaces ◽

Image Position ◽

Invariant Spaces

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] aswhereIf the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

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The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series ◽

10.29235/1561-2430-2021-57-2-156-175 ◽

2021 ◽

Vol 57 (2) ◽

pp. 156-175

Author(s):

P. G. Patseika ◽

Y. A. Rouba

Keyword(s):

Fourier Series ◽

Asymptotic Expression ◽

Conjugate Function ◽

Conjugate Functions ◽

Chebyshev Series ◽

Approximation Properties ◽

Computational Mathematics ◽

Conjugate Fourier Series ◽

Optimal Value ◽

Polynomial Series

Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

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Pointwise approximation of modified conjugate functions by matrix operators of their Fourier series with the use of some parameters

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.51.2020.3080 ◽

2020 ◽

Vol 51 (2) ◽

pp. 145-159

Author(s):

Bogdan Szal ◽

Wlodzimierz Lenski

Keyword(s):

Fourier Series ◽

Periodic Function ◽

Conjugate Function ◽

Conjugate Functions ◽

Pointwise Approximation ◽

Matrix Operators

We extend and generalize the results of Xh. Z. Krasniqi [Acta Comment. Univ.Tartu. Math. 17 (2013), 89-101] and the authors [Acta Comment. Univ. Tartu.Math. 13 (2009), 11-24], [Proc. Estonian Acad. Sci. 2018, 67, 1, 50--60] aswell the jont paper with M. Kubiak [Journal of Inequalities and Applications(2018) 2018:92]. We consider the modified conjugate function $\widetilde{f}%_{r}$ for $2\pi /\rho $--periodic function $f$ . Moreover, the measure ofapproximations depends on \textbf{\ }$\mathbf{\rho }$\textbf{ - }differencesof the entries of matrices defined the method of summability.

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