Orthogonal Trigonometric Approximations Research Articles

Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions

Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\mathbf{B}^r_{p,\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\leq q \leq \infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths.
 For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1<p<2\leq q < \frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\mathbf{B}}^r_{p,\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\theta$.

Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables

We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$.

Estimates of the best orthogonal trigonometric approximations and orthoprojective widths of the classes of periodic functions of many variables in a uniform metric

Some approximative characteristics of classes of periodic functions of many variables $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L∞. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes $$ {L}_{\beta, p}^{\psi }, $$ 1 < p < 1, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.

Estimates of the best orthogonal trigonometric approximations and orthoprojective widths of the classes of periodic functions of many variables in a uniform metric

Some approximative characteristics of classes of periodic functions of many variables L^{\psi}_{\beta, p}, 1 &lt; p &lt; \infty, in a uniform metric are investigated. The first part of the paper is devoted to the construction of estimates of the best orthogonal trigonometric approximations of the mentioned classes in the space L_\infty. In the second part, we have established the ordinal estimates of the orthoprojective widths of the classes L^{\psi}_{\beta, p}, 1 &lt; p &lt; \infty, in the same space, as well as the estimates of another approximative characteristic which is close, in a definite meaning, to the orthoprojective width.

The best M-term trigonometric approximations of the classes of periodic multivariate functions with bounded generalized derivative in the space L q

The order estimates of the best M-term trigonometric approximations of functions \( {D}_{\beta}^{\psi } \) and the classes of (ψ , β)-differentiable periodic multivariate functions in the space L q , 2 ≤ q < ∞ are obtained. It is shown that, under certain conditions imposed on the parameter q; the best M-term trigonometric approximations \( {e}_M{\left({L}_{\beta, 1}^{\psi}\right)}_q \) have better order than the best orthogonal trigonometric approximations \( e\frac{1}{M}{\left({L}_{\beta, 1}^{\psi}\right)}_q \).

Approximating Characteristics of the Classes L β,p ψ of Periodic Functions in the Space L q

We obtain the exact-order estimates of the best m-term and orthogonal trigonometric approximations and establish the order of trigonometric widths for the classes L , in the space L q for some relations between the parameters p and q.

Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of 2π-periodic functions whose (ψ, $$ \beta $$ )-derivatives belong to unit balls in the spaces L p , 1 ≤ p < ∞, in the case where the sequence ψ(k) is such that the product ψ(n)n 1/p may tend to zero slower than any power function and $$ {\displaystyle {\sum}_{k=1}^{\infty }}{\psi}^{p\prime }{(k)}^{p\prime -2}<\infty \kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 1<p<\infty, \frac{1}{p}+\frac{1}{p^{\prime }}=1,\kern0.5em \mathrm{o}\mathrm{r}{\displaystyle {\sum}_{k=1}^{\infty }}\psi (k)<\infty $$ for p = 1. Similar estimates are also established in the Ls-metrics, 1 < s ≤ ∞, for the classes of summable (ψ, $$ \beta $$ )-differentiable functions such that ‖f ‖1 ≤ 1.

Оцінки найкращих ортогональних тригонометричних наближень класів згорток періодичних функцій невеликої гладкості

Оцінки найкращих ортогональних тригонометричних наближень класів згорток періодичних функцій невеликої гладкості

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 orthogonal trigonometric approximations of the classes B Ω p,θ of periodic functions of many variables

We obtain exact-order estimates for the best orthogonal trigonometric approximations of the classes B Ω p,θ of periodic functions of many variables in the space L q .

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B p,θ r ) and Nukol’skii (H p r ) classes of periodic functions of many variables in the metric of L q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L 1 and L ∞ by trigonometric polynomials with the corresponding spectrum.

Best M-term orthogonal trigonometric approximations of the classes ${\boldsymbol{B}}^{\boldsymbol\Omega }_{{\boldsymbol{p},\boldsymbol\Theta }}$ of periodic functions of many variables

We obtain order estimates for the best M-term orthogonal trigonometric approximations of the classes ${{B}}^{\Omega }_{{{p},\Theta }}$ of periodic functions of many variables in the metric of the space L q for 1 < q ≤ p < ∞.

Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric

We study best M-term trigonometric approximations and best orthogonal trigonometric approximations for the classes B and W of periodic functions of several variables in the uniform metric.

Bilinear and trigonometric approximations of periodic functions of several variables of Besov classes

We obtain order-sharp estimates for bilinear approximations of periodic functions of variables of the form , , obtained from functions , , by translating the argument by vectors . We also study the deviations of step hyperbolic Fourier sums on the classes and the best orthogonal trigonometric approximations in , , of functions belonging to these classes.

Best Orthogonal Trigonometric Approximations of Classes of Functions of Many Variables Lβ, p ψ

We obtain order estimates for the best orthogonal trigonometric approximations of classes of functions of many variables Lβ, pψ in the space Lq, 1 2.

On the bestm-term trigonometric and orthogonal trigonometric approximations of functions from the classesL Ψ β,ρ

We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classesL Ψ β,ρ of functions of one variable in the spaceL q in the case 2<p <q < ∞.