?
The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}
Studia Mathematica. 2015. Vol. 231. No. 1. P. 73-81.
The well-known Bohr--Pal theorem
asserts that for every continuous real-valued function f on
the circle T there exists a change of variable, i.e.,
a homeomorphism h of T onto itself, such that the
Fourier series of the superposition foh converges
uniformly. Subsequent improvements of this result imply that
actually there exists a homeomorphism that brings f into the
Sobolev space W_2^{1/2}(T). This refined version of
the Bohr--Pal theorem does not extend to complex-valued
functions. We show that if \alpha<1/2, then there exists a
complex-valued f that satisfies the Lipschitz condition of
order \alpha and at the same time has the property that
foh is not in W_2^{1/2}(T) for every homeomorphism
h of T.
Vladimir Lebedev, / Cornell University. Series math "arxiv.org". 2015. No. 1508.07167.
The well-known Bohr--Pal theorem asserts that for every continuous real-valued function f on the circle T there exists a change of variable, i.e., a homeomorphism h of T onto itself, such that the Fourier series of the superposition f o h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism ...
Added: September 3, 2015
В. В. Лебедев, Известия РАН. Серия математическая 2010 Т. 74 № 2 С. 131-164
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case. ...
Added: April 12, 2012
Vladimir Lebedev, / Cornell University. Series math "arxiv.org". 2015. No. 1508.06673.
We consider the class C(T) of continuous real-valued functions on the circle. For certain classes of functions naturally characterised by the rapidity of decrease of Fourier coefficients we investigate whether it is possible to bring families of functions in C(T) into these classes by a change of variable. This paper was originally published in Matematicheskii ...
Added: September 8, 2015
Ростов н/Д : [б.и.], 2016
The theme of the conference is related to the different areas of mathematics, especially harmonic analysis, functional analysis, operator theory, function theory, differential equations and fractional analysis, developed intensively last decade. The relevance of this topic is related to the study of complex multiparameter objects that require, in particular, to attract operators with variable parameters ...
Added: February 22, 2017
V. V. Lebedev, Sbornik Mathematics 2010 Vol. 201 No. 12 P. 1811-1836
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase. ...
Added: April 12, 2012
V. V. Lebedev, / Cornell University. Series math "arxiv.org". 2016. No. 1603.04539.
It is well--known that certain
properties of continuous functions on the circle $\mathbb T$,
related to the Fourier expansion, can be improved by a change
of variable, i.e., by a homeomorphism of the circle onto
itself. One of the results in this area is the Jurkat--Waterman
theorem on conjugate functions, which improves the classical
Bohr--Pal theorem. In the present work we ...
Added: November 10, 2016
Lebedev V., Olevskii A., / Cornell University. Series math "arxiv.org". 2019. No. arXiv:1803.02177v2.
We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition f oh is in M_p(R^d) for all p, 1 < p < \infty. Moreover, under certain assumptions on a family K of continuous functions, one ...
Added: May 8, 2018
V. V. Lebedev, Functional Analysis and Its Applications 2017 Vol. 51 No. 2 P. 148-151
It is well--known that certain
properties of continuous functions on the circle T,
related to the Fourier expansion, can be improved by a change
of variable, i.e., by a homeomorphism of the circle onto
itself. One of the results in this area is the Jurkat--Waterman
theorem on conjugate functions, which improves the classical
Bohr--P\'al theorem. In the present work we propose ...
Added: June 29, 2017
Lebedev V., Olevskii A., Journal of Mathematical Analysis and Applications 2020 Vol. 481 No. 2 Article 123502
We consider the algebras M_p of
Fourier multipliers and show that for every bounded continuous
function f on R^d there exists a self-homeomorphism
h of R^d such that the superposition foh$ is
in M_p(R^d) for all p, 1<p<\infty. Moreover,
under certain assumptions on a family K of continuous
functions, one h will suffice for all f\in K. A similar
result holds for ...
Added: October 16, 2019
В. В. Лебедев, Функциональный анализ и его приложения 2017 Т. 51 № 2 С. 87-91
It is well--known that certain
properties of continuous functions on the circle T,
related to the Fourier expansion, can be improved by a change
of variable, i.e., by a homeomorphism of the circle onto
itself. One of the results in this area is the Jurkat--Waterman
theorem on conjugate functions, which improves the classical
Bohr--Pal theorem. In the present work we propose ...
Added: June 29, 2017
Vladimir Lebedev, Studia Mathematica 2019 Vol. 247 No. 3 P. 273-283
We consider the Wiener algebra A(T^d) of absolutely convergent Fourier series on the d-torus. For phase functions \phi of a certain special form we obtain lower bounds for the A -norms of e^{i\lambda\varphi} as \lambda tends to \infty. ...
Added: March 23, 2018
V. V. Lebedev, Functional Analysis and Its Applications 2013 Vol. 47 No. 1 P. 27-37
We consider domains D ⊆ ℝn with C1 boundary and study the following question: For what domains D does the Fourier transform 1D of the characteristic function 1D belong to Lp(ℝn)? ...
Added: March 28, 2013
Vladimir Lebedev, / Cornell University. Series math "arxiv.org". 2017. No. arXiv:1611.01739v4.
We concider the Wiener algebra A(T^d) of absolutely convergent Fourier series on the d -torus. For phase functions \phi of a certain special form we obtain lower bounds for the A -norms of e^{i\lambda\phi} as \lambda tends to \infty. ...
Added: November 11, 2016
V. V. Lededev, Izvestiya. Mathematics 2010 Vol. 74 No. 2 P. 347-378
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators in these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case. ...
Added: April 12, 2012
Osipov D., Паршин А. Н., Математический сборник 2020 Т. 211 № 1 С. 125-174
In this work we construct a harmonic analysis on free Abelian groups of rank 2, namely: we construct and investigate spaces of functions and distributions, Fourier transforms and actions of discrete and extended discrete Heisenberg groups. In the case of the rank-2 value group of a two-dimensional local field with finite last residue field we ...
Added: May 20, 2020
V. V. Lebedev, Sbornik Mathematics 2012 Vol. 203 No. 11 P. 1647-1653
We consider the Paley--Wiener spaces of L2-functions whose Fourier transform has a bounded support. We show that every continuous mapping that generates a superposition operator acting on these spaces is affine and injective. ...
Added: January 29, 2013
Chistyakov V., Solycheva O. M., Journal of difference equations and applications 2003 Vol. 9 No. 3-4 P. 407-416
We present necessary and sufficient conditions on the generating functions of operators of substitution (Nemytskii superposition operators), which map the Waterman space λBV of functions of λ-bounded variation on the interval into another space of this type and satisfy the Lipschitz condition. ...
Added: December 4, 2012
Shevgunov T., М. : Издательская группа URSS, 2017
Центральным объектом настоящей книги является метод комплексных амплитуд, позволяющий заменить трудоемкие операции над гармоническими колебаниями на сравнительно простые алгебраические действия, выполняемые над комплексными числами. Метод комплексных амплитуд рассматривается в контексте расчета электрических цепей (линейных инвариантных во времени), токи и напряжения в которых изменяются по гармоническому закону, и последующего анализа полученных результатов. Отдельное внимание в книге уделено задачам расчета ...
Added: April 2, 2019
Goncharuk N. B., Функциональный анализ и его приложения 2012 Т. 46 № 1 С. 13-30
По заданному диффеоморфизму окружности f можно построить отображение, переводящее вещественное число a в число вращения диффеоморфизма f+a. В 1978 г. В. И. Арнольд предложил комплексный аналог этого отображения: каждое число z, Imz>0, переходит в модуль μ(z) эллиптической кривой, которая строится по отображению f+z. В предлагаемой статье исследовано поведение отображения μ вблизи отрезков вещественной оси, на ...
Added: February 18, 2013
Kolesnikov A., / Cornell University. Series math "arxiv.org". 2012. No. 1201.2342.
We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the ...
Added: March 28, 2013
В. В. Лебедев, Функциональный анализ и его приложения 2013 Т. 47 № 1 С. 33-46
We consider domains in Rn with C1 -boundary and study the following question: For what domains does the Fourier transform of the characteristic function delongs to Lp? ...
Added: August 30, 2013
Shevgunov T., М. : ЛЕНАНД, 2014
Центральным объектом настоящей книги является метод комплексных амплитуд, позволяющий заменить трудоемкие операции над гармоническими колебаниями на сравнительно простые алгебраические действия, выполняемые над комплексными числами. Метод комплексных амплитуд рассматривается в контексте расчета электрических цепей (линейных инвариантных во времени), токи и напряжения в которых изменяются по гармоническому закону, и последующего анализа полученных результатов. Отдельное внимание в книге уделено задачам расчета ...
Added: February 7, 2014
В. В. Лебедев, Математический сборник 2012 Т. 203 № 11 С. 121-128
We consider the Paley--Wiener spaces of L2 -functions whose Fourier transform has a bounded support. We show that every continuous mapping that generates a superposition operator acting on these spaces is affine and injective. ...
Added: February 13, 2013
Agranovich M. S., М. : МЦНМО, 2013
Эта книга адресуется математикам, которые занимаются уравнениями в частных производных и функциональным анализом.
Первые две главы содержат вводные курсы. В главе I это теория пространств H s бесселевых потенциалов (s ∈ ; при s 0 это пространства W2 С. Л. Соболева––Л. Н. Слободецкого). В главе II –– теория общих эллиптических уравнений и задач в этих пространствах ...
Added: December 18, 2013