Interrelations between mixed moduli of smoothness in metrics of L
                            p and L
                            ∞

Потапов, М. К.; Симонов, Б. В.

doi:10.3103/s0027132217030044

Cited by 2 publications

(3 citation statements)

References 56 publications

(83 reference statements)

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“…Proof. This lemma is proved as Theorem 4.1 in [7]. To prove property 4, we use the well-known equality (see for example [8], Ch.…”

Section: Auxiliary Statementsmentioning

confidence: 99%

“…Proof. The lemma is proved according to a well-known scheme (see for example [5], [7]). Therefore, we will prove the lemma for m = 2.…”

Section: Now We Will Consider the Functionmentioning

confidence: 99%

“…Definition 3. The mixed modulus of smoothness of a function of order k of the function f ∈ L p,τ (T m ) is determined by the formula (in the case of τ = p, see [7], [8], Ch. 1, sec.…”

Section: Introductionmentioning

confidence: 99%

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Eestimates of Best Approximations of Functions With Logarithmic Smoothness in the Lorentz Space With Anisotropic Norm

Akishev

2020

UMJ

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In this paper, we consider the anisotropic Lorentz space $L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})$ of periodic functions of $m$ variables. The Besov space $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ by trigonometric polynomials under different relations between the parameters $\bar{p}, \bar\theta,$ and $\tau$.The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function $f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})$ to belong to the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$ in the case $1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},$ in terms of the best approximation and prove its unimprovability on the class $E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}$ ${n=0,1,\ldots\},}$ where $E_{n}(f)_{\bar{p},\bar{\theta}}$ is the best approximation of the function $f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})$ by trigonometric polynomials of order $n$ in each variable $x_{j},$ $j=1,\ldots,m,$ and $\lambda=\{\lambda_{n}\}$ is a sequence of positive numbers $\lambda_{n}\downarrow0$ as $n\to+\infty$. In the second section, we establish order-exact estimates for the best approximation of functions from the class $B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}$ in the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$.

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“…Proof. This lemma is proved as Theorem 4.1 in [7]. To prove property 4, we use the well-known equality (see for example [8], Ch.…”

Section: Auxiliary Statementsmentioning

confidence: 99%

“…Proof. The lemma is proved according to a well-known scheme (see for example [5], [7]). Therefore, we will prove the lemma for m = 2.…”

Section: Now We Will Consider the Functionmentioning

confidence: 99%

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Eestimates of Best Approximations of Functions With Logarithmic Smoothness in the Lorentz Space With Anisotropic Norm

Akishev

2020

UMJ

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Refinement of Relations between Mixed Moduli of Smoothness in the Metrics of $$L_p$$ and $$L_\infty$$

Потапов

Симонов

2021

Math Notes

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Interrelations between mixed moduli of smoothness in metrics of L p and L ∞

Cited by 2 publications

References 56 publications

Eestimates of Best Approximations of Functions With Logarithmic Smoothness in the Lorentz Space With Anisotropic Norm

Eestimates of Best Approximations of Functions With Logarithmic Smoothness in the Lorentz Space With Anisotropic Norm

Refinement of Relations between Mixed Moduli of Smoothness in the Metrics of $$L_p$$ and $$L_\infty$$

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