0000000000016189
AUTHOR
Ioan Raşa
Strong Converse Results for Linking Operators and Convex Functions
We consider a family B n , ρ c of operators which is a link between classical Baskakov operators (for ρ = ∞ ) and their genuine Durrmeyer type modification (for ρ = 1 ). First, we prove that for fixed n , c and a fixed convex function f , B n , ρ c f is decreasing with respect to ρ . We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators B n , ρ c applied to convex functions.
Ulam Stability for the Composition of Operators
Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C 0 -semigroup ( T t ) t &ge
Elementary hypergeometric functions, Heun functions, and moments of MKZ operators
We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-Konig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions.
Inequalities for Information Potentials and Entropies
We consider a probability distribution p0(x),p1(x),&hellip
Information potential for some probability density functions
Abstract This paper is related to the information theoretic learning methodology, whose goal is to quantify global scalar descriptors (e.g., entropy) of a given probability density function (PDF). In this context, the core concept is the information potential (IP) S [ s ] ( x ) : = ∫ R p s ( t , x ) d t , s > 0 of a PDF p(t, x) depending on a parameter x; it is naturally related to the Renyi and Tsallis entropies. We present several such PDF, viewed also as kernels of integral operators, for which a precise relation exists between S[2](x) and the variance Var[p(t, x)]. For these PDF we determine explicitly the IP and the Shannon entropy. As an application to Information Theoretic Learning w…
A C0-Semigroup of Ulam Unstable Operators
The Ulam stability of the composition of two Ulam stable operators has been investigated by several authors. Composition of operators is a key concept when speaking about C0-semigroups. Examples of C0-semigroups formed with Ulam stable operators are known. In this paper, we construct a C0-semigroup (Rt)t&ge
Modified Operators Interpolating at Endpoints
Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate at endpoints although they do not preserve the affine functions. We investigate the properties of these modified operators and obtain results concerning iterates and their limits, Voronovskaja-type results and estimates of several differences.
On the composition and decomposition of positive linear operators (VII)
In the present paper we study the compositions of the piecewise linear interpolation operator S?n and the Beta-type operator B?n, namely An:= S?n ?B?n and Gn := B?n ? S?n. Voronovskaya type theorems for the operators An and Gn are proved, substantially improving some corresponding known results. The rate of convergence for the iterates of the operators Gn and An is considered. Some estimates of the differences between An, Gn, Bn and S?n, respectively, are given. Also, we study the behaviour of the operators An on the subspace of C[0,1] consisting of all polygonal functions with nodes {0, 1/2,..., n-1/n,1}. Finally, we propose to the readers a conjecture concerning the eigenvalues of the ope…
Estimates for the differences of positive linear operators and their derivatives
The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Oxur approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, and Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of the first modulus of continuity. In order to analyze the theoretical results in the last section, we consider some numerical examples.