Page 133 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 133
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
[2] C. Bardaro, I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type
series, Jean J. Approx., 6 (2) (2014), 143-165.
[3] P. L. Butzer, A. Fisher, R. L. Stens, Approximation of continuous and discontinuous func-
tions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39.
[4] D. Costarelli, M. Piconi, G. Vinti, On the convergence properties of Durrmeyer-Sampling
type operators in Orlicz spaces, (2020), arXiv: 2007.02450v1, submitted.
[5] D. Costarelli, G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich
Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special issue dedicated to
Prof. Giovanni Prodi, 9 (IV) (2011), 445-468.
[6] J. L. Durrmeyer, Une firmule d’inversion de la transformée de Laplace: applications à la
théorie des moments, Thése de 3e cycle, Universitè de Paris, (1967).
Hölder continuity of the pluricomplex Green function
Rafał Pierzchała, rafal.pierzchala@uj.edu.pl
Jagiellonian University, Poland
I will discuss the following problem of Ples´niak (1988). Let h : U → CN , where U ⊂ CN is
an open set, be a holomorphic map (N, N ∈ N). Assume that a compact set ∅ = K ⊂ CN has
the HCP property (that is, the pluricomplex Green function VK of K is Hölder continuous) and
Kˆ ⊂ U . Under what conditions does it happen that h(K) has the HCP property?
Best Ulam constant of a linear differential operator
Dorian Popa, Popa.Dorian@math.utcluj.ro
Technical University of Cluj-Napoca, Romania
The Ulam stability of an operator L acting in Banach spaces is equivalent with the stability of
the associated equation Lx = y. 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 differential operators.
The linear differential operator with constant coefficients
D(y) = y(n) + a1y(n−1) + . . . + any, y ∈ Cn(R, X)
acting in a Banach space X is Ulam stable if and only if its characteristic equation has no
roots on the imaginary axis. We prove that if the characteristic equation of D has distinct
roots rk satisfying rk > 0, 1 ≤ k ≤ n, then the best Ulam constant of D is KD =
1∞ n
|V | 0 k=1 (−1)k Vk e−rk x dx, where V = V (r1, r2, . . . , rn) and Vk = V (r1, . . . , rk−1,
rk+1, . . . , rn), 1 ≤ k ≤ n, are Vandermonde determinants.
131
[2] C. Bardaro, I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type
series, Jean J. Approx., 6 (2) (2014), 143-165.
[3] P. L. Butzer, A. Fisher, R. L. Stens, Approximation of continuous and discontinuous func-
tions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39.
[4] D. Costarelli, M. Piconi, G. Vinti, On the convergence properties of Durrmeyer-Sampling
type operators in Orlicz spaces, (2020), arXiv: 2007.02450v1, submitted.
[5] D. Costarelli, G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich
Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special issue dedicated to
Prof. Giovanni Prodi, 9 (IV) (2011), 445-468.
[6] J. L. Durrmeyer, Une firmule d’inversion de la transformée de Laplace: applications à la
théorie des moments, Thése de 3e cycle, Universitè de Paris, (1967).
Hölder continuity of the pluricomplex Green function
Rafał Pierzchała, rafal.pierzchala@uj.edu.pl
Jagiellonian University, Poland
I will discuss the following problem of Ples´niak (1988). Let h : U → CN , where U ⊂ CN is
an open set, be a holomorphic map (N, N ∈ N). Assume that a compact set ∅ = K ⊂ CN has
the HCP property (that is, the pluricomplex Green function VK of K is Hölder continuous) and
Kˆ ⊂ U . Under what conditions does it happen that h(K) has the HCP property?
Best Ulam constant of a linear differential operator
Dorian Popa, Popa.Dorian@math.utcluj.ro
Technical University of Cluj-Napoca, Romania
The Ulam stability of an operator L acting in Banach spaces is equivalent with the stability of
the associated equation Lx = y. 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 differential operators.
The linear differential operator with constant coefficients
D(y) = y(n) + a1y(n−1) + . . . + any, y ∈ Cn(R, X)
acting in a Banach space X is Ulam stable if and only if its characteristic equation has no
roots on the imaginary axis. We prove that if the characteristic equation of D has distinct
roots rk satisfying rk > 0, 1 ≤ k ≤ n, then the best Ulam constant of D is KD =
1∞ n
|V | 0 k=1 (−1)k Vk e−rk x dx, where V = V (r1, r2, . . . , rn) and Vk = V (r1, . . . , rk−1,
rk+1, . . . , rn), 1 ≤ k ≤ n, are Vandermonde determinants.
131
