On approximation by Lagrange interpolating polynomials for a subset of the space of continuous functions

S. Zhou

Displaying similar documents to “On approximation by Lagrange interpolating polynomials for a subset of the space of continuous functions”

Ruscheweyh differential operator sets of basic sets of polynomials of several complex variables in hyperelliptical regions.

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A class of analytic functions defined by Ruscheweyh derivative

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The function $f (z) = z^{p} + \sum_{k = 1}^{\infty} a_{p + k} z^{p + k}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_{n, p} (h)$ if ( $D^{n + p} f) / (D^{n + p - 1} f) ≺ h$ , where $D^{n + p - 1} f = (z^{p}) / ({(1 - z)}^{p + n}) * f$ and h is convex univalent in E with h(0) = 1. We study the class $K_{n, p} (h)$ and investigate whether the inclusion relation $K_{n + 1, p} (h) \subseteq K_{n, p} (h)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_{n, p} (a, h)$ of functions satisfying the condition $a * (D^{n + p} f) / (D^{n + p - 1} f) + (1 - a) * (D^{n + p + 1} f) / (D^{n + p} f) ≺ h$ is also studied.

Padé and Hermite-Padé approximation and orthogonality.

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Integral means of certain analytic functions.

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Strong $q$ -variation inequalities for analytic semigroups

Christian Le Merdy, Quanhua Xu (2012)

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Let $T : L^{p} (Ω) \to L^{p} (Ω)$ be a positive contraction, with $1 < p < \infty$ . Assume that $T$ is analytic, that is, there exists a constant $K \geq 0$ such that $∥ T^{n} - T^{n - 1} ∥ \leq K / n$ for any integer $n \geq 1$ . Let $2 < q < \infty$ and let $v^{q}$ be the space of all complex sequences with a finite strong $q$ -variation. We show that for any $x \in L^{p} (Ω)$ , the sequence ${([T^{n} (x)] (λ))}_{n \geq 0}$ belongs to $v^{q}$ for almost every $λ \in Ω$ , with an estimate $∥ {(T^{n} (x))}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ . If we remove the analyticity assumption, we obtain an estimate $∥ {(M_{n} (T) x)}_{n \geq 0} ∥_{L^{p} (v^{q})} \leq C {∥ x ∥}_{p}$ , where $M_{n} (T) = {(n + 1)}^{- 1} \sum_{k = 0}^{n} T^{k}$ denotes the ergodic average of $T$ . We also obtain similar results for strongly continuous semigroups...

Family of analytic functions of complex order.

Frasin, B.A. (2006)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

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