Convergence and rate of approximation in $BV^{\varphi}(\mathbb R^N_+)$ for a class of Mellin integral operators

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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“…On the other hand, the importance of Mellin analysis is well known, not only in approximation theory (we refer to [11,12] for an extensive theory about Mellin operators, while, for other results about homothetic-type and discrete operators in various setting, one can see, e.g. [2,[13][14][15][16][17][18][19][20][21][22]), but also in several other fields, because of its wide applications (see, e.g. [23][24][25][26]).…”

Section: Introductionmentioning

confidence: 99%

Convergence in variation and a characterization of the absolute continuity

Angeloni

Vinti

2015

Integral Transforms and Special Functions

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We study approximation results for a family of Mellin integral operators of the formThe starting point of this study is motivated by the important applications that approximation properties of certain families of integral operators have in image reconstruction and in other fields. In order to treat such problems, to work in BV -spaces in the multidimensional setting of R N + becomes crucial: for this reason we use a multidimensional concept of variation in the sense of Tonelli, adapted from the classical definition to the present setting of R N + equipped with the Haar measure. Using such definition of variation, we obtain a convergence result proving that V [T w f − f ] → 0, as w → +∞, whenever f is an absolutely continuous function; moreover we also study the problem of the rate of approximation. In case of regular kernels, we finally prove a characterization of the absolute continuity in terms of the convergence in variation by means of the Mellin-type operators {T w f } w>0 .

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“…Proof The sufficient part is guaranteed by Theorem 3 in . On the other side, by Theorem ,

T_{w} f \in A C^{φ} (R_{+}^{N})

and hence if, for some

λ > 0

{trueprefixlim}_{w \to + \infty} V^{φ} [λ (T_{w} f - f)] = 0

, then

f \in A C^{φ} (R_{+}^{N})

, by Proposition .

□

…”

Section: A Characterization Of Acφ(r+n)mentioning

confidence: 86%

“…

x_{j}^{'} \in {double-struckR}_{+}^{N - 1}

, and so, using again Proposition , we conclude that

T_{w} f

is φ‐absolutely continuous on I . The proof is complete taking into account that, by Proposition 1 of ,

T_{w} f \in B V^{φ} (R_{+}^{N})

, since

f \in B V^{φ} (R_{+}^{N})

□

…”

Section: A Characterization Of Acφ(r+n)mentioning

confidence: 88%

“…The development of the main properties of such concept are due to Musielak and Orlicz [30], and for this reason its name remained related to the two Polish mathematicians. Further results can be found, for example, in [5]- [7], [15], [24], [29], [33]. We also recall that other concepts of ϕ-variation were given in the literature: among them, for example, the fine ϕ-variation (see, e.g., [1]), the ϕ-variation in the sense of Riesz (see, e.g., [10], [11], [18], [23]) or the ϕ-variation in the sense of Schramm [25], [32].…”

Section: Definitions and Notationsmentioning

confidence: 90%

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