Convergence in variation and a characterization of the absolute continuity

The paper deals with approximation results with respect to the φ‐variation by means of a family of discrete operators for φ‐absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in φ‐variation. The problem of the rate of approximation is also studied. The discrete operators that we consider are deeply connected to some problems of linear prediction from samples in the past, and therefore have important applications in several fields, such as, for example, in speech processing. Moreover such family of operators coincides, in a particular case, with the generalized sampling‐type series on a subset of the space of the φ‐absolutely continuous functions: therefore we are able to obtain a result of convergence in variation also for the generalized sampling‐type series. Some examples are also discussed.

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“…In the main properties of the φ‐variation are presented, while for further papers about such concept we refer to, e.g., .…”

Section: Preliminariesmentioning

confidence: 99%

Discrete operators of sampling type and approximation in ‐variation

Angeloni

Vinti

2017

Mathematische Nachrichten

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“…Remark We point out that, due to the assumptions on the φ‐function φ, we cannot obtain, as particular case of the present frame, results for the multidimensional generalization of the Jordan variation (adapted to the setting of

R_{+}^{N}

equipped with the Haar measure) studied, for example, in and (nonlinear case). We refer to for a discussion about the relationships between

B V

and

B V^{φ}

in the case of the classical Lebesgue measure: analogous considerations hold for the present definitions by means of the logarithmic measure μ.…”

Section: Definitions and Notationsmentioning

confidence: 81%

A concept of absolute continuity and its characterization in terms of convergence in variation

Angeloni

Vinti

2016

Mathematische Nachrichten

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“…The last two situations imply that assumption

(i i i)

holds. From , we have already known that the function

H_{k}

in satisfies assumption

(i v)

.…”

Section: Graphical Illustrations and Numerical Computationsmentioning

confidence: 98%

“…For another application, consider the following special cases: Take

A = F = {[a_{n k}^{(υ)}]}

, the almost convergence method given by . Let

H_{k} false(u false) : = \{\begin{matrix} left u + e^{u / k} - 1, & left if 0 \leq u < 1, \\ left u + e^{1 / false(k u false)} - 1, & left if u \geq 1, \end{matrix}

and the definition of

H_{k} false(u false)

is extended in odd way for

u < 0

(see ). Define 2π‐periodic kernel

L_{k}

, for

t \in [- π, π]

L_{k} false(t false) : = \{\begin{matrix} left 0true \frac{4 k^{2}}{π} \sqrt{0true \frac{1}{k^{2}} - t^{2}}, & left if (||, t) \leq \frac{1}{k} and k = m^{2} (m \in N), \\ left 0true \frac{2 k^{2}}{π} \sqrt{0true \frac{1}{k^{2}} - t^{2}}, & left if (||, t) \leq \frac{1}{k} and k \neq m^{2}, \\ left 0, & left \end{matrix}

…”

Section: Graphical Illustrations and Numerical Computationsmentioning

confidence: 99%

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Approximation by nonlinear integral operators via summability process

Aslan

Duman

2020

Mathematische Nachrichten

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In this paper, we study the approximation properties of nonlinear integral operators of convolution‐type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi‐norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At the end of the paper, we construct a specific sequence of nonlinear operators, which verifies the summability process. Some graphical illustrations and numerical computations are also provided.

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Convergence in variation and a characterization of the absolute continuity

Cited by 21 publications

References 35 publications

Discrete operators of sampling type and approximation in ‐variation

Discrete operators of sampling type and approximation in ‐variation

A concept of absolute continuity and its characterization in terms of convergence in variation

Approximation by nonlinear integral operators via summability process

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