Discrete Approximation Theory

The following sections are included:

• Dedication
• Contents
• Preface

Here we study the approximation properties with rates of generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators. We treat both the unitary and non-unitary cases of these operators. We establish quantitatively the pointwise and uniform convergences of these operators to the unit operator by involving the uniform higher modulus of smoothness of a uniformly continuous function. It follows [3].

Here we give the approximation properties with rates of generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively L_p convergence of these operators to the unit operator by involving the L_p higher modulus of smoothness of an L_p(ℝ) function. It follows [4].

Here we give asymptotic expansions for the generalized discrete versions of unitary Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators. These are of Voronovskaya type expansions and they are connected to the approximation properties of these operators. It follows [3].

In this chapter, we study the fractional generalized smooth discrete singular operators on the real line, the univariate and non-univariate cases, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of associated right and left Caputo fractional derivatives of engaged function. Furthermore, we produce fractional Voronovskaya type results giving the fractional asymptotic expansion of the basic error of our approximation. We give applications and show that our operators are not in general positive. It follows [6].

In this chapter we continue with the study of generalized discrete singular operators over the real line regarding their simultaneous global smoothness preservation property with respect to L_p-norm for 1 ≤ p ≤ ∞, by involving higher order moduli of smoothness. Additionally, we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The Jackson type inequalities that produced in this chapter are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. It follows [4].

In this chapter, we work on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the L_p-norm for 1 ≤ p ≤ ∞. The related established inequalities contain the higher order Lp-modulus of smoothness of the engaged function or its higher order derivative. Also we study the complex-valued fractional generalized discrete singular operators on the real line, regarding their convergence to the unit operator with rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the related function. It follows [5].

Here we establish the uniform approximation properties of multivariate generalized discrete versions of Picard, Gauss. Weierstrass, and Poisson. Cauchy singular operators over ℝ^N, N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We give quantitatively the pointwise and uniform convergences of these operators to the unit operator by involving the multivariate higher order modulus of smoothness. It follows [3].

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss. Weierstrass, and Poisson. Cauchy singular operators over ℝ^N, N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively L_p convergence of these operators to the unit operator by involving the L_p higher modulus of smoothness of an L_p-function. It follows [4].

Here we give asymptotic expansions including simultaneous ones, for the multivariate generalized discrete versions of unitary Picard, Gauss– Weierstrass, and Poisson–Cauchy type singular operators. These are Voronovskaya type expansions and they are connected to the approximation properties of these operators. It follows [4].

In this chapter we study the multivariate generalized discrete singular operators defined on ℝ^N, N ≥ 1, regarding their simultaneous global smoothness preservation property with respect to L_p-norm for 1 ≤ p ≤ ∞, by using higher order moduli of smoothness. Furthermore, we study their simultaneous approximation properties. It follows [7].

In this chapter, we work on the general complex-valued discrete singular operators over ℝ^N, N ≥ 1 and their convergence to the unit operator with rates in the L_p-norm for 1 ≤ p ≤ ∞. The related established inequalities contain the higher order Lp-modulus of smoothness of the engaged function or its higher order derivative. It follows [5].

The following sections is included:

• Index