Interpolation of Sequences (original) (raw)

Theory of Computing Systems / Mathematical Systems Theory, 2010

Naftalevič’s Theorem states that, given a Blaschke sequence, it is possible to modify the arguments of its terms so as to obtain an interpolating sequence. We prove a computable version of this theorem in that it possible, given a Blaschke sequence, to computably modify the arguments of its terms so as to obtain an interpolating sequence. Using this result, we produce a computable, interpolating Blaschke sequence that does not define a computable Blaschke product. This answers a question posed by Matheson and McNicholl in a recent paper. We use Type-Two Effectivity as our foundation.

On the Convergence of (0,1,2) Interpolation

2010

For the Hermite interpolation polynomial, Hm(x) we prove for any function f ∈ C(2q)([−1, 1]) and any s = 0, 1, 2, . . . , q, where q is a fixed integer that |H m (x) − f (x)| = O(1)ω( 1 m , f ) log n n2q−2s . Here m is defined by m = 3n− 1. If f ∈ C(q)([−1, 1]), then |H m − f (x)| = O(1)ω( 1 m , f ) log n (1 − x2)q/2 for x ∈ (−1, 1). 2000 Mathematics Subject Classification: 41A05.

An interpolation theorem

Arkiv för matematik, 1984

Given an interpolation couple (A0, A~), the approximation functional is dcfined

On P ´ Al-Type Interpolation II

In this paper, we study the convergence of Pál-type interpolation on two sets of non-uniformly distributed zeros on the unit circle, which are obtained by projecting vertically the nodes of the real line.

Geometric conditions for interpolation

1983

Let (zn } be a sequence lying in either the upper half-plane or the unit disc in the complex plane. If (zn } is a separated sequence we give a simple geometric condition that implies the sequence is an interpolating sequence for the algebra of bounded holomorphic functions. This result contains most of the known results of this type. 0. Introduction. Let A denote the unit disc I z I < 1 in the complex plane and al its boundary. A sequence {zn} in A is an interpolating sequence for the algebra H?(A) of bounded holomorphic functions in A if, for each bounded sequence of complex numbers {wn4, there exists a function fin Hx(A) such that f(zn) = wn for all n. For {zn} to be interpolating it is necessary and sufficient that

An Interpolation Theorem with Perturbed Continuity

Journal of Functional Analysis, 2002

Let (A 0 , A 1 ) and (B 0 , B 1 ) be two interpolation couples and let T: (A 0 , A 1 ) W (B 0 , B 1 ) be a K-quasilinear operator. The boundedness of the operator from A 0 to B 0 implies K(t, Ta; B 0 , B 1 ) [ M 0 ||a|| A0 and the boundedness of the operator from A 1 to B 1 implies K(t, Ta; B 1 , B 0 ) [ M 1 ||a|| A1 , a ¥ A 0 5 A 1 . We consider perturbations of these two inequalities in the form

AN ALTERNATIVE CHARACTERIZATION OF NORMED INTERPOLATION SPACES BETWEEN ℓ 1 AND ℓ q

arXiv, 2017

Given a constant q ∈ (1, ∞), we study the following property of a normed sequence space E: If {x n } n∈N is an element of E and if {y n } n∈N is an element of ℓ q such that ∞ n=1 |x n | q = ∞ n=1 |y n | q and if the nonincreasing rearrangements of these two sequences satisfy N n=1 |x * n | q ≤ N n=1 |y * n | q for all N ∈ N, then {y n } n∈N ∈ E and {y n } n∈N E ≤ C {x n } n∈N E for some constant C which depends only on E. We show that this property is very close to characterizing the normed interpolation spaces between ℓ 1 and ℓ q. More specificially, we first show that every space which is a normed interpolation space with respect to the couple (ℓ p , ℓ q) for some p ∈ [1, q] has the above mentioned property. Then we show, conversely, that if E has the above mentioned property, and also has the Fatou property, and is contained in ℓ q , then it is a normed interpolation space with respect to the couple ℓ 1 , ℓ q. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in [11].

Interpolation of Sequences (original) (raw)

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