Frontiers in Approximation Theory

The following sections are included:

• Dedication
• Preface
• Contents

Let f ∈ C^p([−1, 1]), p ≥ 0 and let L be a linear left fractional differential operator such that L(f) ≥ 0 throughout [0, 1]. We can find a sequence of polynomials Q_n of degree ≤ n such that L(Q_n) ≥ 0 over [0, 1], furthermore f is approximated uniformly by Q_n. The degree of this restricted approximations is given by an inequalities using the modulus of continuity of f^(p)…

Let f ∈ C^p([−1, 1]), p ≥ 0 and let L be a linear right fractional differential operator such that L(f) ≥ 0 throughout [−1, 0]. We can find a sequence of polynomials Q_n of degree ≤ n such that L(Q_n) ≥ 0 over [−1, 0], furthermore f is approximated uniformly by Q_n. The degree of this restricted approximations is given by an inequalities using the modulus of continuity of f^(p).

Let f ∈ C^r([−1, 1]), r ≥ 0 and let L^* be a linear left fractional differential operator such that L^* (f) ≥ 0 throughout [0, 1]. We can find a sequence of polynomials Q_n of degree ≤ n such that L^* (Q_n) ≥ 0 over [0, 1], furthermore f is approximated left fractionally and simultaneously by Q_n on [−1, 1]. The degree of this restricted approximations is given via inequalities using a higher order modulus of smoothness for f^(r). It follows [4].

Let f ∈ C^r([−1, 1]), r ≥ 0 and let L^* be a linear right fractional differential operator such that L^* (f) ≥ 0 throughout [−1, 0]. We can find a sequence of polynomials Q_n of degree ≤ n such that L^* (Q_n) ≥ 0 over [−1, 0], furthermore f is approximated right fractionally and simultaneously by Q_n on [−1, 1]. The degree of this restricted approximations is given via inequalities using a higher order modulus of smoothness for f^(r). It follows [4].

Let f ∈ C^s([−1, 1]), s ∈ ℕ and L^* be a linear left fractional differential operator such that L^* (f) ≥ 0 on [0, 1]. Then there exists a sequence Q_n, n ∈ ℕ of polynomial splines with equally spaced knots of given fixed order such that L^* (Q_n) ≥ 0 on [0, 1]. Furthermore f is approximated with rates fractionally and simultaneously by Q_n in the uniform norm. This constrained fractional approximation on [−1, 1] is given via inequalities involving a higher modulus of smoothness of f^(s). It follows [2].

Let f ∈ C^s([−1, 1]), s ∈ ℕ and L^* be a linear right fractional differential operator such that L^* (f) ≥ 0 on [−, 0]. Then there exists a sequence Q_n, n ∈ ℕ of polynomial splines with equally spaced knots of given fixed order such that L^* (Q_n) ≥ 0 on [−1, 1]. Furthermore f is approximated with rates fractionally and simultaneously by Q_n in the uniform norm. This constrained fractional approximation on [−1, 1] is given via inequalities involving a higher modulus of smoothness of f^(s). It follows [2].

Here is developed the theory of complete fractional simultaneous monotone uniform polynomial approximation with rates using mixed fractional linear differential operators…

Here is presented the theory of lower order fractional simultaneous monotone uniform polynomial approximation with rates using mixed lower order fractional linear differential operators. To obtain that, we use first ordinary simultaneous polynomial approximation with respect to the highest lower order right and left fractional derivatives of the function under approximation using their moduli of continuity. Then we use the total right and left fractional simultaneous polynomial approximation with rates, as well their convex combination. Based on the last and elegant analytical techniques, we derive preservation of monotonicity by mixed lower order fractional linear differential operators.

Here we study basic approximation properties with rates of our discrete versions of Picard, Gauss-Weierstrass, Poisson-Cauchy singular operators and of two other discrete operators. We prove uniform convergence of these operators to the unit operator. Also all these operators fulfill the global smoothness preservation property. The discussed operators act on the space of uniformly continuous functions over the real line. It follows [2].

Here we study basic approximation properties with rates of our discrete versions of Gauss-Weierstrass and Picard singular operators. We prove quantitatively pointwise and uniform convergence of these operators to the unit operator. Also these operators fulfill the global smoothness preservation property. The discussed operators act on continuous functions over the real line. It follows [4].

Here we introduce some general interpolating neural network operators in the univariate and multivariate cases. Initially we establish the interpolation property of the operators on functions. Then we derive the approximation properties of these operators on functions. We prove first the ordinary real quantitative pointwise and uniform convergences of these operators to the unit. Smoothness of functions is taken into consideration and speed of convergence improves dramatically. As extensions we consider also the fractional, fuzzy, fuzzy-fractional, fuzzy-random, complex and iterated cases. Furthermore we give Voronovskaya type asymptoticexpansions at all studied settings for the errors of related approximations. It follows [24].

Here we start by proving the Riesz representation theorem for positive linear functionals on the space of continuous functions over a time scale. Then we prove further properties for the related Riemann-Stieltjes integral on time scales and we prove the related Hölder's inequality. Next we prove the Hölder's inequality for general positive linear functionals on time scales. We introduce basic concepts of Approximation theory on time scales and we discuss some limitations of the modulus of continuity there. Next we prove the famous Korovkin theorem on time scales, regarding the approximation of unit operator by sequences of positive linear operators on the space of continuity functions defined on a compact interval of a time scale. Then we produce several Shisha-Mond type inequalities related to Korovkin's theorem, putting the convergence of positive linear operators and positive linear functionals in a quantitative form and giving rates of convergence, all operating on Lipschitz functions on a time scale. At the end we present an example of a concrete and genuine positive linear operator on time scales and we give its approximation and interpolation properties over continuous functions. It follows [2].

The following section is included:

• Index