Differentiable Periodic Functions Research Articles

INTEGRATION OF THE NEGATIVE ORDER MODIFIED KORTEWEG–DE VRIES EQUATION WITH A LOADED TERM IN THE CLASS OF PERIODIC FUNCTIONS

Spectral data of the Dirac operator with periodic potential are found. This operator is associated with the negative order modified Korteweg–de Vries equation with a loaded term. The obtained results make it possible to construct a solution to the negative order modified Korteweg–de Vries equation with a loaded term in the class of periodic functions using the inverse spectral problem method. The solvability of the Cauchy problem for an infinite system of Dubrovin–Trubovitz differential equations in the class of three times continuously differentiable periodic functions is proved.

Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

INTEGRATION OF A NONLINEAR HIROTA TYPE EQUATION WITH ADDITIONAL TERMS

In this paper, the inverse spectral problem method is used to integrate a nonlinear Hirota-type equation with additional terms in the class of periodic functions. The evolution of the spectral data of the periodic Dirac operator is introduced, the coefficient of Dirac operator is a solution to the nonlinear Hirota type equation with additional terms. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-zone functions is proved.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.

Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions

In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a π \pi -periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable x x ; and if the number π / 2 \pi /2 is a period (antiperiod) of the initial function, then the number π / 2 \pi /2 is a period (antiperiod) in the variable x x of the solution of the Cauchy problems for the Hirota equation.

THE CAUCHY PROBLEM FOR THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH FINITE DENSITY IN THE CLASS OF PERIODIC FUNCTIONS

In this paper, the method of the inverse spectral problems are used to integrate the nonlinear modified Korteweg-de Vries equation with finite density in the class of periodic functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five-fold continuously differentiable periodic functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies the mKdV equations with finite density.

Estimates of Alexandrov’s n-Width of a Compact Set for Some Infinitely Differentiable Periodic Functions

Estimates of Alexandrov’s n-Width of a Compact Set for Some Infinitely Differentiable Periodic Functions

The Order of Comonotone Approximation of Differentiable Periodic Functions

Let Δ(1)(Y ) be a set of all 2𝜋-periodic functions f that are continuous on the real axis R and change their monotonicity at various fixed points yi ∈ [-𝜋; 𝜋); i = 1; :::; 2s; s ∈ N (i.e., there is a set Y := {yi}i∈ℤ of points yi = yi+2s + 2𝜋 on R such that f are nondecreasing on [yi; yi−1] if i is even, and nonincreasing if i is odd). In the article, a function fY = f 2 C(1) ∩ Δ(1)(Y ) has been constructed suchthat$$ \underset{n\to \infty }{\lim}\sup \frac{n{E}_n^{(1)}(f)}{\omega_4\left({f}^{\prime },\pi /n\right)}=\infty, $$where \( {E}_n^{(1)}(f) \) is the error of the best uniform approximation of the function f ∈ Δ(1)(Y) by trigonometric polynomials of order n ∈ N, which also belong to the set Δ(1)(Y ), and 𝜔4(f’; ∙) is the 4-th modulus of smoothness of the function f’. So, for a certain constant c, the inequality \( {E}_n^{(1)}(f)\le \frac{c}{n}{\omega}_3\left({f}^{\prime },\pi /n\right) \) is the best with respect to the order of the modulus of smoothness.

О наилучшем приближении и значениях поперечников некоторых классов функций в весовом пространстве Бергмана

We consider the extremal problem of finding exact constants in the Jackson-Stechkin type inequalities connecting the best approximations of analytic in the unit circle U={z:|z|<1} functions by algebraic complex polynomials and the averaged values of the higher-order continuity modules of the r-th derivatives of functions in the Bergman weight space B_2,γ. The classes of analytic in the unit circle functions W^((r))_m(τ) and W^((r))_m(τ,Φ), which satisfy some specific conditions are introduced. For the introduced classes of functions, the exact values of some known n-widths are calculated. In this paper, we use the methods of solving extremal problems in normalized spaces of functions analytic in a circle and a wellknown method developed by V.M. Tikhomirov for estimating from below the n-widths of functional classes in various Banach spaces. The results obtained in the work generalize and extend the results of the works by S.B. Vakarchuk and A.N. Shchitova obtained for the classes of differentiable periodic functions to the case of analytic in the unit circle functions belonging to the Bergmann weight space.

The order of comonotone approximation of differentiable periodic functions

Let $\Dely$ be a set of all $2\pi$-periodic functions $f$ that are continuous on the real axis $R$\ and\ change their monotonicity at various fixed points $y_{i}\in\lbrack-\pi,\pi),\ i=1,...,2s,\ s\in N$ (i.e., there is a set $Y:=\{y_{i}\}_{i\in\mathbb{Z}}$ of points $y_{i}=y_{i+2s}+2\pi$ on $R$ such that $f$ are nondecreasing on $[y_{i},y_{i-1}]$ if $i$ is even, and nonincreasing if $i$ is odd). In the article, a function $f_{Y}=f\in C^{(1)}\cap\Dely$ has been constructed such that \[ \lim_{n\rightarrow\infty}\sup\frac{n\,E_{n}^{(1)}(f)}{\omega_{4}(f^{\prime },\pi/n)}=\infty, \] where $E_{n}^{(1)}(f)$ is the error of the best uniform approximation of the function $f\in\Dely$ by trigonometric polynomials of order $n\in N$, which also belong to the set $\Dely$, and $\omega_{4}(f^{\prime},\cdot)$ is the $4$-th modulus of smoothness of the function $f^{\prime}.$ So, for a certain constant $c$, the inequality $E_{n}^{(1)}(f)\leq\frac{c}{n}\omega _{3}(f^{\prime},\pi/n)$ is the best with respect to the order of the modulus of smoothness.

О точных значениях поперечников некоторых классов функций из L_2

In this paper we find sharp inequalities of Jackson-Stechkin type between the best approximations of periodic differentiable functions by trigonometric polynomials and generalized moduli of continuity of m-th order in the space L_2. The exact values of various n-widths of classes of functions from L_2 defined by the generalized modus of continuity of the $r$-th derivative of the function f are calculated.

Точные равенства наближения функций класса Соболева их обобщенными интегралами Пуассона

Решение задач о движении системы взаимодействующих материальных точек в большинстве случаев сводится как к обычным дифференциальным уравнениям, так и к уравнениям в частных производных. Одним из решений такого типа уравнений являются так называемые обобщенные интегралы Пуассона, которые в отдельных случаях превращаются в хорошо известные интегралы Абеля-Пуассона или бигармонические интегралы Пуассона. Существует ряд результатов по приближению различных классов дифференцируемых периодических и непериодических функций вышеупомянутыми интегралами (так называемая задача Колмогорова-Никольского в терминологии А.И. Степанця). Но практически во всех решенных задачах Колмогорова-Никольского как для интегралов Абеля-Пуассона, так и для бигармонических интегралов Пуассона с точки зрения математического моделирования (вычислительного эксперимента) существенный недостаток. Суть этого недостатка состоит в том, что в большинстве решенных ранее задач Колмо-горова-Никольского для интегралов Абеля-Пуассона и бигармонических интегралов Пуассона (в конечном итоге) был получен только главный и остаточный члены приближения, что существенно может влиять на точность вычислительного опыта. Данная работа посвящена получению чётких равенств приближения функций классов Соболева их обобщенными интегралами Пуассона. Итак, доказанная в работе теорема является обобщением и уточнением ранее известных результатов, характеризующих апроксимативные свойства интегралов Абеля-Пуассона и бигармонических интегралов Пуассона на классах дифференцируемых периодических функций. Особенностью решенной в работе задачи приближения для обобщенного интеграла Пуас-сона на классах дифференцирующих функций является то, что полученный результат удалось записать с помощью известных констант Ахиезера-Крейна-Фавара. Указанный факт значительно повышает точность результата математического моделирования (вычислительного эксперимента) любого реального процесса, описываемого с помощью обобщенного интеграла Пуассона. Эти результаты в дальнейшем смогут значительно расширить рамки применения задач Колмогорова-Никольского к математическому моделированию.

On optimal interval quadrature formulae on classes of differentiable periodic functions

We solved the problem about the best interval quadrature formula on the class $W^r F$ of differentiable periodic functions with arbitrary permutation-invariant set $F$ of derivatives of order $r$. We proved that the formula with equal coefficients and $n$ node intervals, which have equidistant middle points, is the best on given class.

Non-Differentiability of the Convolution of Differentiable Real Functions

We provide an example of two \(2\)-periodic everywhere differentiable functions \(f,g\colon\mathbb{R}\to\mathbb{R}\) whose convolution \(f*g\) fails to be differentiable at every point of some perfect (thus, uncountable) set \(P\subset\mathbb{R}\). This shows that the convolution operator can actually destroy the differentiability of these maps, rather than introducing additional smoothness (as it is usually the case). New directions and open problems are also posed.

Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

ATOMIC FUNCTIONS AND THEIR GENERALIZATIONS IN DATA PRO-CESSING: FUNCTION THEORY APPROACH

Theory of atomic functions, which are solutions with a compact support of the linear functional differential equations with a constant coefficients and linear transforms of the argument, was created in the 70's of the 20th century because of the necessity to solve different applied problems, in particular, boundary value problems. One of the reasons for the appearance of atomic functions and some other classes of functions was the inability to use such classic approximation tools as algebraic and trigonometric polynomials. V.A. Rvachev up-function is the most famous and widely used atomic function. With the passage of time and the development of technologies, the existing problems are changing and fundamentally new problems appear. For instance, now the big data processing is one of the most important problems. It should be mentioned that the suitable mathematical tools must be used to obtain the desired result. This paper is devoted to fundamentals of applications of some atomic functions and their generalizations to data processing and lossy information compression. In this paper we consider the main properties of these functions from the function theory point of view and give their interpretation with respect to information processing. Smoothness, compact support and good approximation properties are the main advantages of atomic functions. Moreover, the spaces of atomic functions and the spaces of generalized Fup-functions, which are the natural generalization of V.A. Rvachev Fup-functions, are asymptotically extremal for approximation of periodic differentiable functions. This means that in the terms of A.N. Kolmogorov width these functions are just as effective as classic trigonometric polynomials {1, cos(nx), sin(nx)}. Hence, the replacement of discrete transforms based on trigonometric functions on similar transforms based on atomic functions and generalized Fup-functions is quite promising. For this purpose we introduce discrete atomic transform and generalized discrete atomic transform. We also discuss the dependence of data processing results on order of smoothness and size of support of the used functions. Theoretical justification of the application of some atomic functions and generalized Fup-functions to data processing and, in particular, lossy data compression is the main result of this paper

Sharp Remez-type Inequalities of Various Metrics in the Classes of Functions with Given Comparison Function

For any p ϵ [1,∞], 𝜔 > 0, β ϵ (0, 2𝜔), and any measurable set B ⊂ I d := [0,d], μB ≤ β, we establish a sharp Remez-type inequality of various metrics $$ {E}_0{(x)}_{\infty}\le \frac{{\left\Vert \varphi \right\Vert}_{\infty }}{E_0{{{\left(\varphi \right)}_L}_p}_{\left({I}_{2\omega}\backslash {B}_1\right)}}{\left\Vert x\right\Vert}_{Lp\;\left({I}_d\backslash B\right)} $$ in the classes S𝜑(ω) of d-periodic (d ≥ 2ω) functions x with a given sine-shaped 2ω-periodic comparison function 𝜑, where B1 := [(ω − β)/2, (ω+β)/2] and E0(f)Lp(G) is the best approximation of the function f by constants in the metric of the space Lp(G). In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.

Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces

Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces

Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let $${{\Bbb S}_{B,n}}$$ be the space of functions s of the form $$s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} $$ . This paper describes all spaces $${{\Bbb S}_{B,n}}$$ that satisfy the exact inequality $$E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}$$ . (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

Sharp Remez-Type Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

We prove a sharp Remez-type inequality of different metrics $$ {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left\{\frac{{\left\Vert x\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash B\right)}}{{\left\Vert {\varphi}_r\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash {B}_1\right)}}\right\}}^{\alpha }{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern0.5em q>p>0,\kern0.5em \alpha =\left(r+1/q\right)/\left(r+1/p\right), $$ for 2π -periodic functions x ∈ $$ {L}_{\infty}^r $$ satisfying the condition where $$ L{(x)}_p\coloneq \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:\left[a,b\right]\subset \left[0,2\uppi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\} $$ , B ⊂ [0, 2π], μB ≤ β/λ (λ is chosen so that $$ {\left\Vert x\right\Vert}_p={\left\Vert {\varphi}_{\uplambda, r}\right\Vert}_{L_p\left[0,2\uppi /\uplambda \right]} $$ ), φ r is the ideal Euler’s spline of order r, and $$ {B}_1\coloneq \left[\frac{-\uppi -\beta /2}{2},\frac{-\uppi +\beta /2}{2}\right]\cup \left[\frac{\uppi -\beta /2}{2},\frac{\uppi +\beta /2}{2}\right] $$ . As a special case, we establish sharp Remez-type inequalities of different metrics for trigonometric polynomials and polynomial splines satisfying the condition (*).