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APPROXIMATION THEORY AND APPLICATIONS (MS-78)
Variation diminishing type estimates for generalized sampling operators
and applications
Laura Angeloni, laura.angeloni@unipg.it
Department of Mathematics and Computer Science, University of Perugia, Italy
Coauthors: Danilo Costarelli, Gianluca Vinti
The variation diminishing estimate is a classical result that is usually investigated working in
BV spaces with some classes of operators: such result essentially ensures that the variation of
the operator is not bigger than the variation of the function to which it is applied. We will present
estimates of this kind, besides results about convergence in variation, for multivariate sampling-
type operators. Differently from the one-dimensional frame, where variation diminishing type
results are usually quite easy to be achieved, the multidimensional case is more delicate: nev-
ertheless it is interesting, also from an applicative point of view, since it is connected to some
problems of Digital Image Processing, in particular to smoothing procedures.
References
[1] L. Angeloni, G. Vinti, Estimates in variation for multivariate sampling-type operators,
Dolomites Research Notes on Approximation, 14(2) (2021), 1–9.
[2] L. Angeloni, D. Costarelli, G. Vinti, A characterization of the convergence in variation for
the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755–767.
[3] L. Angeloni, D. Costarelli, G. Vinti, Convergence in variation for the multidimensional
generalized sampling series and applications to smoothing for digital image processing,
Ann. Acad. Sci. Fenn. Math., 45 (2020), 751–770.
[4] L. Angeloni, D. Costarelli, M. Seracini, G. Vinti, L. Zampogni, Variation diminishing-
type properties for multivariate sampling Kantorovich operators, Boll. Unione Mat. Ital.,
Special Issue "Measure, Integration and Applications" dedicated to Prof. Domenico Can-
deloro, 13 (2020), 595–605.
[5] P.L. Butzer, A. Fisher, R.L. Stens, Generalized sampling approximation of multivariate
signals: theory and applications, Note Mat., 1 (10), 173–191 (1990).
[6] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Prediction by Samples From the Past With
Error Estimates Covering Discontinuous Signals, IEEE Trans. Inform. Theory, 56 (1),
614–633 (2010).
Best Ulam constant of a linear difference equation
Alina Ramona Baias, baias.alina@math.utcluj.ro
Technical University of Cluj Napoca, Romania
An equation is called Ulam stable if for every approximate solution of it there exists an exact
solution near it. We present some results on Ulam stability for some linear difference equations.
In a Banach space X the linear difference equation with constant coefficients xn+p =
a1xn+p−1 + . . . + apxn, is Ulam stable if and only if the roots rk, 1 ≤ k ≤ p, of its characteristic
equation do not belong to the unit circle. If |rk| > 1, 1 ≤ k ≤ p, we prove that the best Ulam
124
Variation diminishing type estimates for generalized sampling operators
and applications
Laura Angeloni, laura.angeloni@unipg.it
Department of Mathematics and Computer Science, University of Perugia, Italy
Coauthors: Danilo Costarelli, Gianluca Vinti
The variation diminishing estimate is a classical result that is usually investigated working in
BV spaces with some classes of operators: such result essentially ensures that the variation of
the operator is not bigger than the variation of the function to which it is applied. We will present
estimates of this kind, besides results about convergence in variation, for multivariate sampling-
type operators. Differently from the one-dimensional frame, where variation diminishing type
results are usually quite easy to be achieved, the multidimensional case is more delicate: nev-
ertheless it is interesting, also from an applicative point of view, since it is connected to some
problems of Digital Image Processing, in particular to smoothing procedures.
References
[1] L. Angeloni, G. Vinti, Estimates in variation for multivariate sampling-type operators,
Dolomites Research Notes on Approximation, 14(2) (2021), 1–9.
[2] L. Angeloni, D. Costarelli, G. Vinti, A characterization of the convergence in variation for
the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755–767.
[3] L. Angeloni, D. Costarelli, G. Vinti, Convergence in variation for the multidimensional
generalized sampling series and applications to smoothing for digital image processing,
Ann. Acad. Sci. Fenn. Math., 45 (2020), 751–770.
[4] L. Angeloni, D. Costarelli, M. Seracini, G. Vinti, L. Zampogni, Variation diminishing-
type properties for multivariate sampling Kantorovich operators, Boll. Unione Mat. Ital.,
Special Issue "Measure, Integration and Applications" dedicated to Prof. Domenico Can-
deloro, 13 (2020), 595–605.
[5] P.L. Butzer, A. Fisher, R.L. Stens, Generalized sampling approximation of multivariate
signals: theory and applications, Note Mat., 1 (10), 173–191 (1990).
[6] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Prediction by Samples From the Past With
Error Estimates Covering Discontinuous Signals, IEEE Trans. Inform. Theory, 56 (1),
614–633 (2010).
Best Ulam constant of a linear difference equation
Alina Ramona Baias, baias.alina@math.utcluj.ro
Technical University of Cluj Napoca, Romania
An equation is called Ulam stable if for every approximate solution of it there exists an exact
solution near it. We present some results on Ulam stability for some linear difference equations.
In a Banach space X the linear difference equation with constant coefficients xn+p =
a1xn+p−1 + . . . + apxn, is Ulam stable if and only if the roots rk, 1 ≤ k ≤ p, of its characteristic
equation do not belong to the unit circle. If |rk| > 1, 1 ≤ k ≤ p, we prove that the best Ulam
124
