Periodic Functions Of Several Variables Research Articles

Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain the exact-order estimates of approximation of the Nikol'skii-Besov-type classes $B^{\Omega}_{\infty,\theta}$ of periodic functions of several variables with a given function $\Omega(t)$ of a special form by using linear operators satisfying certain conditions. The approximation error is estimated in the metric of the space $L_{\infty}$. The obtained estimates of the considered approximation characteristic, in addition to independent interest, can be used to establish the lower bounds of the orthowidths of the corresponding functional classes.

On Universal Sampling Recovery in the Uniform Norm

It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol’skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions of two variables, where the Fibonacci point sets are used for recovery.

Characteristics of the linear and nonlinear approximations of the Nikol’skii–Besov-type classes of periodic functions of several variables

Characteristics of the linear and nonlinear approximations of the Nikol’skii–Besov-type classes of periodic functions of several variables

Characteristics of the linear and nonlinear approximations of the Nikol’skii-Besov-type classes of periodic functions of several variables

Estimates that are accurate by order of magnitude have been obtained for some characteristics of the linear and nonlinear approximations of the isotropic classes of the Nikol'skii--Besov-type \textit{$\mathbf{B}% ^{\,\omega}_{p,\theta}$} of periodic functions of several variables in the spaces $B_{q,1}, 1 \leq q \leq \infty$. A specific feature of those spaces, as linear subspaces of $L_q$, is that the norm in them is ``stronger'' than the $L_q$-norm.

ON ESTIMATES OF M-TERM APPROXIMATIONS ON CLASSES OF FUNCTIONS WITH BOUNDED MIXED DERIVATIVE IN THE LORENTZ SPACE

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space $$L_{q, \tau }(\mathrm {T}^{m})$$ , the class of functions with bounded mixed fractional derivative $$W_{q, \tau }^{\overline{r}}$$ , $$1< q, \tau < \infty$$ , and studies the order of the best M-term approximation of a function $$f \in L_{p, \tau }(\mathrm {T}^{m})$$ by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class $$W_{q, \tau _{1}}^{\overline{r}}$$ in the norm of the space $$L_{p, \tau _{2 }}(\mathrm {T}^{m})$$ for various relations between the parameters $$p, q, \tau _{1}, \tau _{2}$$ .

Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

In this paper, we continue the study of approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p&lt;q&lt;\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.

Estimates of the best approximations of the functions of the Nikol’skii–Besov class in the generalized space of Lorentz

In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. Exact in order estimates of the best approximation by trigonometric polynomials of functions of the Nikol’skii–Besov class are obtained.

Estimates of approximative characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables with given majorant of mixed moduli of continuity in the space $L_{q}$

In this paper, we continue the study of approximative characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p&lt;q&lt;\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.

Estimates of approximative characteristics of the classes B p , θ Ω $$ {B}_{p,\theta}^{\Omega} $$ of periodic functions of many variables with given majorant of mixed continuity moduli in the space L ∞

We obtain exact-order estimates of the approximation of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of several variables in the space L ∞ , by using operators of orthogonal projection, as well as linear operators subjected to some conditions.

Estimates for the Kolmogorov widths of the Nikol’skii–Besov–Amanov classes in the Lorentz space

The Lorentz space with anisotropic norm of periodic functions of several variables is considered. Estimates for the Kolmogorov widths of the Nikol’skii–Besov–Amanov classes in the Lorentz space with anisotropic norm are found.

Estimation of the Best Linear Approximations for the Classes B p , θ r $$ {B}_{\mathrm{p},\theta}^{\mathrm{r}} $$ and Singular Numbers of the Integral Operators

We establish the exact-order estimates for the best bilinear approximations of the Nikol’skii–Besov classes $$ {B}_{\mathrm{p},\theta}^{\mathrm{r}} $$ of periodic functions of several variables. We also determine the orders of singular numbers for the integral operators with kernels from the classes $$ {B}_{\mathrm{p},\theta}^{\mathrm{r}} $$ .

Widths of the Anisotropic Besov Classes of Periodic Functions of Several Variables

We establish the exact-order estimates for the Kolmogorov and orthoprojective widths of anisotropic Besov classes of periodic functions of several variables in the spaces Lq.

Mixed generalized modulus of smoothness and approximation by the “angle” of trigonometric polynomials

The notion of general mixed modulus of smoothness of periodic functions of several variables in the spaces Lp is introduced. The proposed construction is, on the one hand, a natural generalization of the general modulus of smoothness in the one-dimensional case, which was introduced in a paper of the first author and in which the coefficients of the values of a given function at the nodes of a uniform lattice are the Fourier coefficients of a 2π-periodic function called the generator of the modulus; while, on the other hand, this construction is a generalization of classical mixed moduli of smoothness and of mixed moduli of arbitrary positive order. For the modulus introduced in the paper, in the case 1 ≤ p ≤ +∞, the direct and inverse theorems on the approximation by the “angle” of trigonometric polynomials are proved. The previous estimates of such type are obtained as direct consequences of general results, new mixed moduli are constructed, and a universal structural description of classes of functions whose best approximation by “angle” have a certain order of convergence to zero is given.

Approximations of the classes MB p,θ r of periodic functions of several variables by polynomials according to the Haar system

Some problems of approximation of the classes MBp,θr of periodic functions of several variables by polynomials constructed by the Haar system are investigated. For the functions of these classes, the exact-order estimates of the best m-term approximation by polynomials according to the Haar system and the upper bounds of approximations by step-hyperbolic Fourier–Haar sums are obtained.

Trigonometric Approximations and Kolmogorov Widths of Anisotropic Besov Classes of Periodic Functions of Several Variables

We describe the Besov anisotropic spaces of periodic functions of several variables in terms of the decomposition representation and establish the exact-order estimates of the Kolmogorov widths and trigonometric approximations of functions from unit balls of these spaces in the spaces L q .

Approximating Characteristics of the Analogs of Besov Classes with Logarithmic Smoothness

We obtain the exact-order estimates of some approximating characteristics for the analogs of Besov classes of periodic functions of several variables (with logarithmic smoothness).

Estimates of the approximation characteristics of the classes B p,θ Ω of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain order-sharp estimates of the orthogonal projection widths of the classes Bp,θΩ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.

Best [formula omitted]-term trigonometric approximation of periodic functions of several variables from Nikol’skii–Besov classes for small smoothness

We find the asymptotic orders of best m-term trigonometric approximation for Nikol’skii–Besov classes Bp,θr of periodic functions of several variables in the space Lq, 1≤p≤2<q<∞, for the cases of “small” smoothness dp−dq<r<dp and for the “critical” value of the smoothness r=dp. These results are complementary to the estimates of the best m-term trigonometric approximation for classes Bp,θr obtained by R.A. DeVore and V.N. Temlyakov.

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes Bp,θr of periodic functions of several variables in the space Lq are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x−y), x, y ∈ \(\mathbb{T}\)d = Πj=1d[−π, π], f(x) ∈ Bp,θr, by linear combinations of products of functions of d variables are established.

Best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $ by trigonometric polynomials with the “numbers” of harmonics from graded hyperbolic crosses in the metric of the space L q :