On Equivalence of Moduli of Smoothness

Hu, Yingkang

doi:10.1006/jath.1997.3264

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…Inequalities (7.4) were proved in [29,Lemma 2.1] (see also [27]) in the case 1 ≤ p < ∞. It is easy to see that the same also holds in the case 0 < p < 1.…”

Section: Smoothness Of Functions Versus Smoothness Of Approximation Pmentioning

confidence: 70%

Smoothness of functions versus smoothness of approximation processes

Kolomoitsev

Tikhonov

2020

Bull. Math. Sci.

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We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

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“…Inequalities (7.4) were proved in [29,Lemma 2.1] (see also [27]) in the case 1 ≤ p < ∞. It is easy to see that the same also holds in the case 0 < p < 1.…”

Section: Smoothness Of Functions Versus Smoothness Of Approximation Pmentioning

confidence: 70%

Smoothness of functions versus smoothness of approximation processes

Kolomoitsev

Tikhonov

2020

Bull. Math. Sci.

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“…Then (19), (20), and (22) yield (18). Now let us consider the problem concerning the sharp order of decrease of the best approximation in the spaces H r,α w,p .…”

Section: Properties Of the Best Approximation Direct And Inverse Thementioning

confidence: 99%

On approximation of functions by algebraic polynomials in Hölder spaces

Kolomoitsev

Lomako

Prestin

2016

Mathematische Nachrichten

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Abstract. We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.

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“…However, for some functions f ¥ C [a, b] , one has t k w m − k (f (k) , t) [ Cw m (f, t) (1) for m \ k, where C > 0 is some constant independent of t for small t. This kind of works began from a result of Yu and Zhou [5], and was investigated by Hu [2] and Hu and Yu [3]. As a whole, all these results indicate that for splines with arbitrary (fixed) knots, the inequality (1) holds in general L p spaces for small t. The present paper will investigate polynomials for which the inequality (1) holds.…”

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confidence: 96%

Some Remarks on Equivalence of Moduli of Smoothness

Zhou

2001

Journal of Approximation Theory

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The present paper investigates polynomials for which the inverse inequality for moduli of smoothness holds. The technique for approach is different from the previous works for splines and is elegantly organized. © 2001 Elsevier Science Key Words: equivalence; modulus of smoothness; polynomial. a, b] ), and w k (f, t) [a, b] be the modulus of smoothness of order k of f ¥ C [a, b] , as usual. We will write w(f, t)=w(f, t) [a, b] for convenience if there is no confusion. Let f(x) be a continuous function on the interval [a, b] which hasIt is well known that a, b] , where w 0 (f, t)=||f|| [a, b] The inverse result of the above inequality does not hold in general. However, for some functions f ¥ C [a, b] , one has t k w m − k (f (k) , t) [ Cw m (f, t) (1)

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On Equivalence of Moduli of Smoothness

Cited by 8 publications

References 14 publications

Smoothness of functions versus smoothness of approximation processes

Smoothness of functions versus smoothness of approximation processes

On approximation of functions by algebraic polynomials in Hölder spaces

Some Remarks on Equivalence of Moduli of Smoothness

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