On some extensions of Bernstein's inequality for trigonometric polynomials

Runovski, K.; Schmeißer, Hans-Jürgen

doi:10.7169/facm/1538186723

Cited by 31 publications

(15 citation statements)

References 1 publication

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“…150-151], [15,Theorem 3.2] gives sufficient conditions for the validity of (5.3). Recall, p = min(1, p).…”

Section: Inequalities For Trigonometric Polynomialsmentioning

confidence: 99%

On Approximation Methods Generated by Bochner-Riesz Kernels

Runovski¹,

Schmeißer²

2008

J Fourier Anal Appl

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Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved.Keywords Bochner-Riesz kernels and means · Families of operators · Necessary and sufficient conditions of convergence · K-functional · Realizations and moduli of smoothness related to the Laplacian Mathematics Subject Classification (2000) 42A10 · 42A15

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“…150-151], [15,Theorem 3.2] gives sufficient conditions for the validity of (5.3). Recall, p = min(1, p).…”

Section: Inequalities For Trigonometric Polynomialsmentioning

confidence: 99%

On Approximation Methods Generated by Bochner-Riesz Kernels

Runovski¹,

Schmeißer²

2008

J Fourier Anal Appl

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“…This follows from (Theorem 4.1, [10]) where it has been proved that the Fourier transform of ψη for an infinitely differentiable (defined on R d \{0}) homogeneous function ψ of order α > 0, which is not polynomial, belongs to the space L p (R d ) if and only if p > d/(d + α).…”

Section: Introductionmentioning

confidence: 90%

“…Proofs of (i)-(iii) can be found in [10] (Theorems 3.1 and 3.2) and [11]. For (ii) we also refer to [13], pp.…”

Section: Operators and Inequalities Of Fourier Multiplier-typementioning

confidence: 99%

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Moduli of Smoothness Related to the Laplace-Operator

Runovski

Schmeißer

2014

J Fourier Anal Appl

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We introduce and study a series of new moduli of smoothness in the multivariate case in L p -spaces of periodic functions. The main focus lies on the case 0 < p < 1. We prove a direct Jackson-type estimate and provide necessary and sufficient conditions with respect to the dimension d and to integrability p for the equivalence of these moduli and polynomial K -functionals related to the Laplaceoperator. As a consequence we obtain an inverse Bernstein-type estimate. Moreover, we are able to characterize the approximation error in case of approximation by families of linear polynomial operators which are generated by Bochner-Riesz kernels in terms of the introduced moduli.Keywords Trigonometric approximation · Fourier multipliers · Moduli of smoothness · K -functionals · Jackson-and Bernstein-type theorems · Bochner-Riesz means and families Mathematics Subject Classification 42A10 · 42A15 · 42B08 · 42B15 · 46E35 Communicated by

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“…Note that the operator is well defined because the function F −1 (D(ρ)N M j ) is summable on R d , see, e.g., [31]. Let us give several important examples of the Weyl-type operators:…”

Section: The Case Of Weak Compatibility Of ϕ and ϕmentioning

confidence: 99%

Approximation by periodic multivariate quasi-projection operators

Kolomoitsev

Krivoshein

Skopina

2020

Journal of Mathematical Analysis and Applications

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Approximation properties of quasi-projection operators Qj(f, ϕ, ϕ) are studied. Such an operator is associated with a function ϕ satisfying the Strang-Fix conditions and a tempered distribution ϕ such that compatibility conditions with ϕ hold. Error estimates in the uniform norm are obtained for a wide class of quasi-projection operators defined on the space of uniformly continuous functions and on the anisotropic Besov spaces. Under additional assumptions on ϕ and ϕ, two-sided estimates in terms of realizations of the K-functional are also obtained.

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On some extensions of Bernstein's inequality for trigonometric polynomials

Cited by 31 publications

References 1 publication

On Approximation Methods Generated by Bochner-Riesz Kernels

On Approximation Methods Generated by Bochner-Riesz Kernels

Moduli of Smoothness Related to the Laplace-Operator

Approximation by periodic multivariate quasi-projection operators

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