General Moduli of Smoothness and Approximation by Families of Linear Polynomial Operators

Runovski, K.; Schmeißer, Hans-Jürgen

doi:10.1007/978-3-319-08801-3_11

Cited by 19 publications

(20 citation statements)

References 17 publications

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“…The next lemma gives an analogue of inequality (2.14) for the realizations of K-functional. [32]). Let f ∈ L p (T), 0 < p ≤ ∞, and α > 0.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…where β ∈ N ∪ (1/p 1 − 1, ∞), p 1 = min(p, 1) (see [4] for the case p ≥ 1 and [32] for the case 0 < p < 1). Inequality (2.14) implies that the optimal rate of decrease of the modulus…”

Section: 2mentioning

confidence: 99%

“…For the first time, the modulus (1.8) appeared in 1970's (see [4, p. 788] and [39]). At present, fractional moduli of smoothness are extensively studied and have several important applications to sharp inequalities of different metrics and embedding theorems (see [18], [32], [33], [35], [36], [43], see also the monograph [27] and the literature therein). It is known that for any f ∈ L p (T), 1 ≤ p < ∞, and α, β > 0, the following two inequalities are fulfilled:…”

Section: Introductionmentioning

confidence: 99%

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Inequalities in Approximation Theory Involving Fractional Smoothness in L p, 0 < p < 1

Kolomoitsev

Lomako

2019

Topics in Classical and Modern Analysis

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In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in Lp, 0 < p < 1. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness ω β (f (α) , δ)p, where f (α) is a fractional derivative of the function f , are derived. A description of the class of functions with the optimal rate of decrease of a fractional modulus of smoothness is given.2000 Mathematics Subject Classification. 42A10, 26A33, 41A17, 41A25, 41A28.

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“…The next lemma gives an analogue of inequality (2.14) for the realizations of K-functional. [32]). Let f ∈ L p (T), 0 < p ≤ ∞, and α > 0.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inequalities in Approximation Theory Involving Fractional Smoothness in L p, 0 < p < 1

Kolomoitsev

Lomako

2019

Topics in Classical and Modern Analysis

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“…В [10] получена эквивалентность ошибки приближения семействами линейных полиномиальных операторов, обобщенного -функционала (см. [8]) и общего модуля гладкости при условии эквивалентности их генераторов.…”

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“…то мы получаем эквивалентность ошибки приближения средними Фурье модулю гладкости нечетного порядка [8], [9]. Доказательства эквивалентности генераторов могут быть найдены, например, в [8]- [10]. Заметим, что при построении конкретных примеров наибольшую трудность представляет проверка условия Θ 2 ∈ .…”

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