K. I. Oskolkov

K. I. Oskolkov
University of South Carolina | USC · Department of Mathematics

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55
Publications
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596
Citations

Publications

Publications (55)
Article
For the function \(H:\mathbb{R}^2 \mapsto \mathbb{C}\), \(H: = (p.v.)\sum\nolimits_{n \in \mathbb{Z}\backslash \{ 0\} } {\tfrac{{\exp \left\{ {\pi i\left( {tn^2 + 2xn} \right)} \right\}}} {{2\pi in}}}\) of two real variables (t, x) ∈ ℝ2, we study the uniform moduli of continuity and the variations of the restrictions H|t and H|x onto the lines para...
Article
The function ψ:= ∑n∈ℤbsol;{0} eΠi(tn 2+2xn)/(πin2), {t, x} ∈ ℝ2, is studied as a (generalized) solution of the Cauchy initial value problem for the Schrödinger equation. The real part of the restriction of ψ on the line x = 0, that is, the function, t ∈ ℝ, was suggested by B. Riemann as a plausible example of a continuous but nowhere differentiable...
Article
In this paper, we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ƒ(x) in the metric {ie129-03}, where {ie129-04} is the unit ball |x| ≤ 1 in the plane ℝ2. The notation we use is the following: {fx129-01}. Here {ie...
Article
In this paper, estimates of the rate of convergence almost everywhere of the Fourier series of a continuous function and its conjugate are obtained. These estimates, expressed in terms of its best approximation functions and its modulus of continuity, cannot be strengthened in a number of cases.Bibliography: 9 items.
Article
This paper studies the approximation of functions of two variables by functions whose graphs in 3-space are continuous surfaces which are piecewise flat. By analogy with the case of functions of one variable, we naturally call such approximating functions polygonal. Bibliography: 8 titles.
Article
Full-text available
Streszczenie We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. $\sqrt{\delta}\,$-families of initial functions are cons...
Article
In this paper, we prove a multiple analog of the theorem proved by Arkhipov and the author in 1987, which provides an estimate for the discrete Hilbert transform with polynomial phase. For the linear case, the corresponding estimates of the sum of multiple trigonometric series was proved by Telyakovskii.
Article
It is proved that the double sin-series with the hyperbolic phase S(x):=∑ m=1 ∞ ∑ n=1 ∞ sinmnx m 2 +n 2 is convergent for all real x, and that S(x) is bounded as a function of x. The value of the infinite double sum is understood as the common value of the limits of the sequences of the partial sums over expanding families of coordinate-wise convex...
Article
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Let P be a bi-variate algebraic polynomial of degree n with the real senior part, and Y = {yj}n1 an n-element collection of pairwise non- colinear unit vectors on the real plane. It is proved that there exists a rigid rotation Y ' of Y by an angle ' = '(P,Y ) ∈ (0,�/n) such that P equals the sum of n plane wave polynomials, that propagate in the di...
Article
Full-text available
The goal is to compare free (non-linear), equispaced ridge and algebraic polynomial approximations R fr N [f ]; R eq N [f ]; EN [f ] of individual functions f(x) in the norm of L 2 (IB 2 ), IB 2 -- the unit disc jxj 1 on the plane IR 2 . By definition R fr N [f ] := inf R2W fr N kf Gamma Rk; R eq N [f ] := min R2W eq N kf Gamma Rk; EN [f ] := min P...
Article
Full-text available
Let$$h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $$ (\((i = \sqrt { - 1;} t,x\)-real variables). It is proved that in the rectangle\(D: = \left\{ {(t,x):...
Article
A simple and explicit construction of an orthnormal trigonometric polynomial basis in the spaceC of continuous periodic functions is presented. It consists simply of periodizing a well-known wavelet on the real line which is orthonormal and has compactly supported Fourier transform. Trigonometric polynomials resulting from this approach have optima...
Article
The present paper is a survey of the author’s recent research in the one-dimensional trigonometric series of the type$$\sum\limits_n {\hat f\left( n \right)} {e^{2\pi i\left( {{n^r}{x_r} + \cdots + n{x_1}} \right)}}.$$ (1.1)
Article
The following special function of two real variables x 2 and x 1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤ 1. In particular, it is proved that for each fixed x 2 and uniformly in X 2 the function H(x 2 , x 1 ) is of weakly bounded 2-variation in the variable x 1 over the period [0, 1]...
Article
Properties of convergence and uniform boundedness are studied for partial sums of a special trigonometric series with an algebraic polynomial in the exponent of the imaginary exponential.Bibliography: 9 titles.
Article
In the present paper the problem of optimal linear quadrature formulae is treated. Classes of continuous periodic functions are considered. These classes are defined by restrictions imposed on the action of linear differential operators with constant coefficients. More general classes of convolutions are discussed, as well. The problem in question...
Article
Estimates are obtained of the rate of approximation almost everywhere as a function of the modulus of continuity of the approximated functions in , and of the set from which the approximating functions are chosen. From this point of view the author studies the approximation of functions by Steklov means, partial sums of Fourier-Haar series, arbitra...
Article
Letf be a continuous periodic function with Fourier sums Sn(f), and let En(f)=En be the best approximation tof by trigonometric polynomials of order n. The following estimate is proved: ||f - Sn (f)|| \leqslant cåv = n2u \fracEv v - n + 1 .||f - S_n (f)|| \leqslant c\sum\nolimits_{v = n}^{2u} {\frac{{E_v }}{{v - n + 1}}} . (Here c is an absolute c...
Article
We prove the existence of a functionf(t), which is continuous on the interval [0, 1], is of bounded variation, minf(t)=0, maxf(t)=1, for which the integral $$I(x) = \frac{1}{\pi }\int_0^\infty {\left[ {\int_0^1 {cos} y(f(l) - x)\left| {df(l)} \right|} \right]} dy$$ diverges for almost all X ? [0, 1]. This result gives a negative answer to a questio...
Article
It is proved that if the continuous periodic functionf has bounded || f - Sn (f) || \leqslant cò0w(pn - 1 ) log(vF (f) \mathord/ \vphantom (f) F F(x))dx.\left\| {f - S_n (f)} \right\| \leqslant c\int_0^{\omega (\pi n^{ - 1} )} {\log (v_\Phi } {{(f)} \mathord{\left/ {\vphantom {{(f)} \Phi }} \right. \kern-\nulldelimiterspace} \Phi }(\xi ))d\xi . H...
Article
It is proved that for each modulus of continuity \omega (\delta) in the class H_{\omega} there exists a function f such that for any increasing sequence \{n_i\}_{i=1}^{\infty} of natural numbers there is a point x at which \displaystyle \varlimsup_{t \to \infty} \frac{S_{n_{i}}(f,x)-f(x)}{\omega (n_i^{-1}) \log{n_i}} \geqslant A > 0, \displaystyle...
Article
Necessary and sufficient conditions are derived for the convergence of a trigonometric series to a function of bounded variation on the interval (a, ß) ?[-p, p]. For the case in which the coefficients satisfy certain conditions, the continuity of the sum function is investigated.

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