Page 129 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 129
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
Discretisation of integrals on compact spaces using distance functions
Martin Buhmann, buhmann@math.uni-giessen.de
Justus-Liebig University, Germany
For the purpose of partitioning compact sets, discretisation of integrals and finding quadrature
rules on compact sets, it is important to have estimates for the ensuing error of the approxima-
tion. It is desirable to have estimates on the remainders that are independent of space dimension,
and of course we wish the errors to decrease as fast as possible when the number of summands
in the discretisation increases. In this joint work with Feng Dai and Yeli Niu (Edmonton) we
find such error estimates using regular partitions with particular attention to (but not only to)
discretisations on spheres. In fact, our estimates are quite general, they apply to compact path-
connected metric spaces, and we are able to improve several earlier error estimates from the
literature.
Integral-type operators on mobile intervals
Mirella Cappelletti Montano, mirella.cappellettimontano@uniba.it
Università degli Studi di Bari Aldo Moro, Italy
Coauthor: Vita Leonessa
In this talk, we present a sequence (Cn)n≥1 of positive linear operators, introduced in [1] and
acting on spaces of continuous functions as well as on spaces of integrable functions on [0, 1].
These operators represent a Kantorovich-type modification, on mobile intervals, of the ones
discussed in [2].
We state some qualitative properties of the sequence (Cn)n≥1 and we prove that it is an
approximation process both in C([0, 1]) and in Lp([0, 1]), also providing some estimates of the
rate of convergence. Moreover, we determine an asymptotic formula and we prove that suitable
iterates of the operators Cn converge, both in C([0, 1]) and, under suitable assumptions, in
Lp([0, 1]) to a limit semigroup. Finally, we compare our operators with other existing ones in
the literature showing that they allow a lower approximating error estimate.
References
[1] M. Cappelletti Montano, Vita Leonessa, On a sequence of Kantorovich-type operators,
Constr. Math. Anal. 2 (3) (2019), 130-143.
[2] D. Cardenas-Morales, P. Garrancho, I. Ras¸a, Bernstein-type operators which preserve
polynomials, Comput. Math. Appl. 62 (1) (2011), 158–163.
Kernel-based approximation methods on graphs
Wolfgang Erb, wolfgang.erb@unipd.it
University of Padova, Italy
We study how the concept of positive definite functions can be transfered to a graph setting in
order to approximate graph signals with generalized shifts of a graph basis function (GBF). This
concept merges kernel-based approximation with spectral theory on graphs and can be regarded
127
Discretisation of integrals on compact spaces using distance functions
Martin Buhmann, buhmann@math.uni-giessen.de
Justus-Liebig University, Germany
For the purpose of partitioning compact sets, discretisation of integrals and finding quadrature
rules on compact sets, it is important to have estimates for the ensuing error of the approxima-
tion. It is desirable to have estimates on the remainders that are independent of space dimension,
and of course we wish the errors to decrease as fast as possible when the number of summands
in the discretisation increases. In this joint work with Feng Dai and Yeli Niu (Edmonton) we
find such error estimates using regular partitions with particular attention to (but not only to)
discretisations on spheres. In fact, our estimates are quite general, they apply to compact path-
connected metric spaces, and we are able to improve several earlier error estimates from the
literature.
Integral-type operators on mobile intervals
Mirella Cappelletti Montano, mirella.cappellettimontano@uniba.it
Università degli Studi di Bari Aldo Moro, Italy
Coauthor: Vita Leonessa
In this talk, we present a sequence (Cn)n≥1 of positive linear operators, introduced in [1] and
acting on spaces of continuous functions as well as on spaces of integrable functions on [0, 1].
These operators represent a Kantorovich-type modification, on mobile intervals, of the ones
discussed in [2].
We state some qualitative properties of the sequence (Cn)n≥1 and we prove that it is an
approximation process both in C([0, 1]) and in Lp([0, 1]), also providing some estimates of the
rate of convergence. Moreover, we determine an asymptotic formula and we prove that suitable
iterates of the operators Cn converge, both in C([0, 1]) and, under suitable assumptions, in
Lp([0, 1]) to a limit semigroup. Finally, we compare our operators with other existing ones in
the literature showing that they allow a lower approximating error estimate.
References
[1] M. Cappelletti Montano, Vita Leonessa, On a sequence of Kantorovich-type operators,
Constr. Math. Anal. 2 (3) (2019), 130-143.
[2] D. Cardenas-Morales, P. Garrancho, I. Ras¸a, Bernstein-type operators which preserve
polynomials, Comput. Math. Appl. 62 (1) (2011), 158–163.
Kernel-based approximation methods on graphs
Wolfgang Erb, wolfgang.erb@unipd.it
University of Padova, Italy
We study how the concept of positive definite functions can be transfered to a graph setting in
order to approximate graph signals with generalized shifts of a graph basis function (GBF). This
concept merges kernel-based approximation with spectral theory on graphs and can be regarded
127
