An Improved General Stochastic Inequality
Approximation by perturbed neural network operators
This article deals with the determination of the rate of convergence to the unit of each of three newly introduced perturbed normalized neural network operators of one hidden layer. These are given through the modulus of continuity of the function involved or its high order derivative that appears in the right-hand side of the associated Jackson type inequalities. The activation function is very general, in particular it can derive from any sigmoid or bell-shaped function. The right-hand sides of...
Left general fractional monotone approximation theory
We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative over I,...
General uniform approximation theory by multivariate singular integral operators
We study the uniform approximation properties of general multivariate singular integral operators on , N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators to which this theory can be applied directly.
Ostrowski inequalities for cosine and sine operator functions
Rate of convergence of some neural network operators to the unit-univariate case, revisited
On some shift invariant integral operators, univariate case
In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...
Representation formulae for (C₀) m-parameter operator semigroups
Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.
On global smoothness preservation in complex approximation
By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.
Multidimensional Opial inequalities for functions vanishing at an interior point
In this paper we generalize Opial inequalities in the multidimensional case over balls. The inequalities carry weights and are proved to be sharp. The functions under consideration vanish at the center of the ball.
Differentiated shift-invariant multivariate integral operators.
A discrete stochastic Korovkin theorem.
Ostrowski Type Inequalities over Spherical Shells
2000 Mathematics Subject Classification: 26D10, 26D15. Here are presented Ostrowski type inequalities over spherical shells. These regard sharp or close to sharp estimates to the difference of the average of a multivariate function from its value at a point.
Most General Fractional Representation Formula for Functions and Implications
Here we present the most general fractional representation formulae for a function in terms of the most general fractional integral operators due to S. Kalla, [3], [4], [5]. The last include most of the well-known fractional integrals such as of Riemann-Liouville, Erdélyi-Kober and Saigo, etc. Based on these we derive very general fractional Ostrowski type inequalities. 2010 Mathematics Subject Classification: 26A33, 26D10, 26D15.
Geometric and approximation properties of some singular integrals in the unit disk.
Fractional Korovkin Theory Based on Statistical Convergence
2000 Mathematics Subject Classification: 41A25, 41A36, 40G15. In this paper, we obtain some statistical Korovkin-type approximation theorems including fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
Fractional Trigonometric Korovkin Theory in Statistical Sense
2000 Mathematics Subject Classification: 41A25, 41A36. In the present paper, we improve the classical trigonometric Korovkin theory by using the concept of statistical convergence from the summability theory and also by considering the fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
Page 1