Page 132 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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APPROXIMATION THEORY AND APPLICATIONS (MS-78)
[2] T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., Voronovskaya type results for
Bernstein-Chlodovsky operators preserving e−2x, J. Math. Anal. Appl. 491 (2020), no.
1, 124307, 14 pp.
Functional differential equations with maxima, via step by step
contraction principle
Diana Otrocol, Diana.Otrocol@math.utcluj.ro
Technical University of Cluj-Napoca, Romania
Coauthor: Veronica Ilea
T.A. Burton presented in some examples of integral equations a notion of progressive con-
tractions on C([a, ∞[). In 2019, I.A. Rus formalized this notion (I.A. Rus, Some variants of
contraction principle in the case of operators with Volterra property: step by step contraction
principle, Advances in the Theory of Nonlinear Analysis and its Applications 3 (2019) No. 3,
111-120), put "step by step" instead of "progressive" in this notion, and give some variant of
step by step contraction principle in the case of operators with Volterra property on C([a, b], B)
and C([a, ∞[, B), where B is a Banach space. In this paper we use the abstract result given by
I.A. Rus, to study some classes of functional differential equations with maxima.
Approximation by Durrmeyer-Sampling Type Operators in Functional
Spaces
Michele Piconi, michele.piconi@unifi.it
Department of Mathematics and Computer Science, University of Perugia,
1, Via Vanvitelli, 06123 Perugia, Italy
Coauthors: Gianluca Vinti, Danilo Costarelli
Sampling-type operators have been introduced in order to give an approximate version of the
celebrated classical sampling theorem. Among these operators, we have studied the Durrmeyer-
Sampling type operators (DSO) [4], (see also [6, 2]), which represent a further generalization
of the well-known Generalized [3] and Kantorovich-Sampling operators [1, 5].
In order to follow a unifying approach, we have provided a general result in terms of conver-
gence, which is represented by a modular convergence theorem in Orlicz spaces. From the latter
result, the convergence in Lp-spaces follows as particular case. This approximation result for
DSO is important, in some particular case, from the applications point of view, e.g. in image
processing, where we have to work with not-necessarily continuous signals.
For the sake of completeness of the theory, we have also studied the continuous case, providing
a pointwise and uniform convergence theorem and quantitative estimates.
Moreover, all the above convergence results for DSO can also be extended in the multidimen-
sional setting.
References
[1] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Kantorovich-Type Generalized Sampling
Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing,
9 (6) (2007), 29-52.
130
[2] T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., Voronovskaya type results for
Bernstein-Chlodovsky operators preserving e−2x, J. Math. Anal. Appl. 491 (2020), no.
1, 124307, 14 pp.
Functional differential equations with maxima, via step by step
contraction principle
Diana Otrocol, Diana.Otrocol@math.utcluj.ro
Technical University of Cluj-Napoca, Romania
Coauthor: Veronica Ilea
T.A. Burton presented in some examples of integral equations a notion of progressive con-
tractions on C([a, ∞[). In 2019, I.A. Rus formalized this notion (I.A. Rus, Some variants of
contraction principle in the case of operators with Volterra property: step by step contraction
principle, Advances in the Theory of Nonlinear Analysis and its Applications 3 (2019) No. 3,
111-120), put "step by step" instead of "progressive" in this notion, and give some variant of
step by step contraction principle in the case of operators with Volterra property on C([a, b], B)
and C([a, ∞[, B), where B is a Banach space. In this paper we use the abstract result given by
I.A. Rus, to study some classes of functional differential equations with maxima.
Approximation by Durrmeyer-Sampling Type Operators in Functional
Spaces
Michele Piconi, michele.piconi@unifi.it
Department of Mathematics and Computer Science, University of Perugia,
1, Via Vanvitelli, 06123 Perugia, Italy
Coauthors: Gianluca Vinti, Danilo Costarelli
Sampling-type operators have been introduced in order to give an approximate version of the
celebrated classical sampling theorem. Among these operators, we have studied the Durrmeyer-
Sampling type operators (DSO) [4], (see also [6, 2]), which represent a further generalization
of the well-known Generalized [3] and Kantorovich-Sampling operators [1, 5].
In order to follow a unifying approach, we have provided a general result in terms of conver-
gence, which is represented by a modular convergence theorem in Orlicz spaces. From the latter
result, the convergence in Lp-spaces follows as particular case. This approximation result for
DSO is important, in some particular case, from the applications point of view, e.g. in image
processing, where we have to work with not-necessarily continuous signals.
For the sake of completeness of the theory, we have also studied the continuous case, providing
a pointwise and uniform convergence theorem and quantitative estimates.
Moreover, all the above convergence results for DSO can also be extended in the multidimen-
sional setting.
References
[1] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Kantorovich-Type Generalized Sampling
Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing,
9 (6) (2007), 29-52.
130
