Directional K- Functionals Claudia Cottin

Cottin, Claudia

doi:10.1007/bf03322331

Cited by 1 publication

(6 citation statements)

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“…we know that the definition of A£ is independent of the specific choice of Y in (2). Moreover, we obtain from Lemma 3.2 and Lemma 3.3 that A*g € for all g € ^(IF 1 ).…”

Section: Applicationsmentioning

confidence: 90%

“…In this case, ||7 -Α-γ|| < (1 + π) • ω(η; β) for all 7 € 0 2π (Κ} with β := supj. A [sin 2 (cf., e.g., [4]). Hence we obtain where /?,· = sup^ Ai [sin 2 ( : "5 £ ) ; xj.…”

Section: Applicationsmentioning

confidence: 99%

“…in (6), which yields a somewhat different result than repetition of Brought to you by | University of Pennsylvania Authenticated Download Date | 6/16/15 3:27 AM Also estimates in terms of Ä'-functionals may be transferred by a procedure similar to the one in this section. For the definition of appropriate directional Κ-Junctionals, which are equivalent to the moduli (6), see [2].…”

Section: Lemma 43 Let a : C C Be An Operator Such There Exist C ε mentioning

confidence: 99%

“…with ξ &Y. We decompose any χ G R 1 uniquely as χ = y + s£ with y € Y and s 6 R. Given a univariate (not necessarily linear) operator A : C^ilR) -• C^R) its d-variate directional extension with respect to Y and ξ € Ζ* \ Y is defined via (2) Λ^ΟΦ^Ο,+ ·£);*] for g € Cjjrii? '), k € W0, where χ = y + s£ € R 1 , and η = g(y + ·ξ) is the univariate function given by 7(5) = g(y + s£), 3 6 R. (We equivalently use the notation Af(s) and Α[η\ s].…”

Section: Definition Of Directional Extension Operatorsmentioning

confidence: 99%

“…For example, we applied directional extensions of Steklov means (see Example 2.1 (ii)) in [2] as smoothing functions for general ξ € R? \ {0}.…”

Section: Applicationsmentioning

confidence: 99%

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Directional Extensions of Univariate Operators and Boolean Sum Approximation

Cottin¹

1993

Analysis

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Many approximation processes for multivariate real-valued functions are constructed on the basis of parametric extensions of univariate approximation operators. A parametric extension of a univariate operator with respect to the i/-th component is defined in such a way that it acts on a multivariate function as if all variables except of the i/-th one were fixed. We generalize this approach by introducing directional extensions of univariate operators, which are defined similarly as parametric extensions, but with respect to some arbitrary direction in ϋ Λ . In particular, we investigate approximation by Boolean sums of directional extension operators.

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“…we know that the definition of A£ is independent of the specific choice of Y in (2). Moreover, we obtain from Lemma 3.2 and Lemma 3.3 that A*g € for all g € ^(IF 1 ).…”

Section: Applicationsmentioning

confidence: 90%

“…In this case, ||7 -Α-γ|| < (1 + π) • ω(η; β) for all 7 € 0 2π (Κ} with β := supj. A [sin 2 (cf., e.g., [4]). Hence we obtain where /?,· = sup^ Ai [sin 2 ( : "5 £ ) ; xj.…”

Section: Applicationsmentioning

confidence: 99%