Departamento de Matemáticas
URI permanente para esta comunidadhttps://hdl.handle.net/10953/44
En esta Comunidad se recogen los documentos generados por el Departamento de Matemáticas y que cumplen los requisitos de Copyright para su difusión en acceso abierto.
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Examinando Departamento de Matemáticas por Autor "Adell, José A."
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Ítem Asymptotic and Non-asymptotic Results in the Approximation by Bernstein Polynomials(Birkhäuser - Springer, 2022-06-29) Adell, José A.; Cárdenas-Morales, DanielThis paper deals with the approximation of functions by the classical Bernstein polynomials in terms of the Ditzian–Totik modulus of smoothness. Asymptotic and non-asymptotic results are respectively stated for continuous and twice continuously differentiable functions. By using a probabilistic approach, known results are either completed or strengthened.Ítem Estimates in direct inequalities for the Szász–Mirakyan operator(Springer, 2022-12-16) Adell, José A.; Cárdenas-Morales, DanielThis paper deals with the approximation of continuous functions by the classical Szász– Mirakyan operator. We give new bounds for the constant in front of the second order Ditzian–Totik modulus of smoothness in direct inequalities. Asymptotic and non asymptotic results are stated. We use both analytical and probabilistic methods, the latter involving the representation of the operators in terms of the standard Poisson process. A smoothing technique based on a modification of the Steklov means is also applied.Ítem On the uniqueness conjecture for the maximum Stirling numbers of the second kind(Springer, 2021-04-15) Adell, José A.; Cárdenas-Morales, DanielThe Stirling numbers of the second kind S(n,k) satisfy S(n,0) < · · · < S(n,kn) ≥ S(n,kn+1) > · · · > S(n,n). A long standing conjecture asserts that there exists no n ≥ 3 such that S(n,kn) = S(n,kn +1). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that S(n,kn)> S(n,kn+1).Ítem Random Linear Operators Arising from Piecewise Linear Interpolation on the Unit Interval(Birkhäuser - Springer, 2022-09-09) Adell, José A.; Cárdenas-Morales, DanielWe introduce a sequence of random linear operators arising from piecewise linear interpolation at a set of random nodes on the unit interval. We show that such operators uniformly converge in probability to the target function, providing at the same time rates of convergence. Analogous results are shown for their deterministic counterparts, derived by taking expectations of the aforementioned random operators. Special attention is paid to the case in which the random nodes are the uniform order statistics, where an explicit form for their associated deterministic operators is provided. This allows us to compare the speed of convergence of the aforementioned operators with that of the random and deterministic Bernstein polynomialsÍtem Stochastic Bernstein polynomials: uniform convergence in probability with rates(Springer, 2020-02-27) Adell, José A.; Cárdenas-Morales, DanielWe introduce stochastic variants of the classical Bernstein polynomials associated with a continuous function f , built up from a general triangular array of random variables. We discuss the uniform convergence in probability of the approximation process that they represent, providing at the same time rates of convergence. In the particular case in which the triangular array of random variables consists of the uniform order statistics, we give a positive answer to a conjectured raised in Wu and Zhou (Adv. Comput. Math. 46, 8, 2020) about an exponential rate of convergence in probability.