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R. M. Trigub

Publications and source records attributed to R. M. Trigub.

7 recordsLinked to original sources
arXiv2019

On the best approximation of constants by polynomials with integer coefficients

In this paper, exact rate of decrease of best approximations of non-integer numbers by polynomials with integer coefficients of the growing exponentials is found on a disk in complex plane, on a cube in $\mathbb{R}^d$, and on a ball in $\mathbb{R}^d$. While in the first two cases the $\sup$-norm is used, the third one is fulfilled in $L_p$, $1\leq p<\infty$. Comments are also given (two remarks in the end of the paper).

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arXiv2017

On multiply monotone functions

In this paper, the algebra of the differences of two multiply monotone functions on $\mathbb{R}_+=(0,+\infty)$ is studied. A sufficient condition for the function $f_0\big(|x|_{p,d}\big)$, where $|x|_{p,d}=\Big(\sum\limits_{j=1}^d|x_j|^p\Big)^{\frac{1}{p}}$, $p\in(0,+\infty]$, to be represented as the Fourier transform is given.

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arXiv2016

On various moduli of smoothness and $K$-functionals

In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $§1$. In $\S2$ functions on the line $\mathbb{R}$ are studied. A typical (well-known) result is as follows: for each $2π$-periodic function in $L_p$ on the period, for any $p\in[1,+\infty]$ ($L_\infty=C$) and $r\in\mathbb{N}$, there is a trigonometric polynomial $τ_{r,n}(f)$ of degree not greater than $n$ such that \big\|f-τ_{r,n}(f)\big\|_p\asympω_r\Big(f;\frac{1}{n}\Big)_p\asymp \inf\limits_{g}\Big\{\|f-g\|_p+\frac{1}{n^r}\big\|g^{(r)}\big\|_p\Big\}, where the positive constants in these bilateral inequalities depend only on $r$. In $§3$ we deal with functions on $\mathbb{R}^d$ ($d\geq2$), while in $§4$ with functions on Banach spaces. The paper is partially of survey nature. The proofs are given only for Theorems 2.2, 3.9 and those in $§4$. Related open problems are formulated in $§5$. The list of references contains 52 items.

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arXiv2015

On the Fourier transform of function of two variables which depend only on the maximum of these variables

For functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from $L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform $\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well as of a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geq t^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$ to $L_{1}(\mathbb{R}^{1}_{+})$. As for the positivity of $\widehat{f}$ on $\mathbb{R}^{2}$, it is completely reduced to the same question on $\mathbb{R}^{1}$ for a function $f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.

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arXiv2015

Almost everywhere summability of Fourier series with indicating the set of convergence

The following problem is studied in this paper: Which multipliers $\{λ_{k, n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-π, π]$ $$ \sum_{k=-\infty}^\infty λ_{k, n}\hat{f}_k e^{ikx}, $$ where $\widehat{f}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function $\int_0^x f$ exists. A criterion for the convergence of the $(C, 1)$-means ($λ_{k, n}=(1-\frac {|k|}{n+1})_+$) is found, while in the general case $λ_{k, n}=ϕ(\frac {k}{n+1})$ a sufficient condition is derived for the convergence at all such points (that is, almost everywhere). The answer is given in terms of the belonging of $ϕ(x)$ and $xϕ'(x)$ to the Wiener algebra of absolutely convergent Fourier integrals. The obtained results are supplemented by some examples.

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arXiv1996

Some Topics in Fourier Analysis and Approximation Theory

This manuscript presents shortly the results obtained by participants of the scientific seminar which is held more than twenty years under leadership of the author at Donetsk University. In the list of references main publications are given. These results are published in serious scientific journals and reported at various conferences, including international ones at Moscow,ICM66; Kaluga,1975; Kiev,1983; Haifa,1994; Zürich,ICM94; Moscow,1995. The area of investigation is the Fourier analysis and the theory of approximation of functions. Used are methods of classical analysis including special functions, Banach spaces, etc., of harmonic analysis in finitedimensional Euclidean space, of Diophantine analysis, of random choice, etc. The results due to the author and active participants of the seminar, namely E. S. Belinskii, O. I. Kuznetsova, E. R. Liflyand, Yu. L. Nosenko, V. A. Glukhov, V. P. Zastavny, Val. V. Volchkov, V. O. Leontyev, and others, are given. Besides the participants of the seminar and other mathematicians from Donetsk, many mathematicians from other places were speakers at the seminar, in particular, A.A. Privalov, Z.A. Chanturia, Yu.A. Brudnyi, N.Ya. Krugljak, V.N. Temlyakov, B.D. Kotlyar, A.N. Podkorytov, M.A. Skopina, A.A. Ligun, A.S. Romanyuk, V.A. Martirosyan. Besides the papers of the participants of the seminar, only monographs and survey papers are given in the list of references. The most part of this preprint will appear soon in extended form in the author's book "Fourier Analysis and Approximation Theory", World Federation Publishers, Inc., P.O.Box 48654, Atlanta, GA 30362-0654, USA.

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