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About 359 papers for “interpolating sequences”

Interpolating sequences

Nicola Arcozzi, Richard Rochberg, Eric T. Sawyer et al. · Mathematical surveys and monographs · 2019

In this paper, we investigate the interpolating question in QK spaces. Using some skills of complex analysis and harmonic analysis, we obtain a necessary and sufficient condition for a sequence {zn} in the unit disc to be an interpolating sequence for the analytic function space QK∩H∞, which extends part of results in Qp spaces.

Cited by 0

Lipschitz interpolating sequences

Jiménez-Vargas, A., Zoca, Abraham Rueda · ArXiv.org · 2025

Let $X$ be a metric space with a base point $0$, and let $\mathrm{Lip}_0(X)$ be the Banach space of all Lipschitz functions $f:X\longrightarrow \mathbb R$ such that $f(0)=0$. Given a set of points $\left((x_i,y_i)\right)_{i\in I}$ in $X^2$ with $x_i\neq y_i$ for all $i\in I$, we study the following interpolation problem: when for each bounded set $\left(\alpha_i\right)_{i\in I}$ in $\mathbb{R}$ the algorithm $$ \frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\alpha_i\qquad (i\in I) $$ can be implemented by a function $f\in\mathrm{Lip}_0(X)$? Our approach involves the concept of a Beurling set of functions in $\mathrm{Lip}_0(X)$ for $\left((x_i,y_i)\right)_{i\in I}$ which has shown to be useful in the so-called transportation problem.

Cited by 0Open access

Recursive interpolating sequences

Francesc Tugores · Open Mathematics · 2018

Abstract This paper is devoted to pose several interpolation problems on the open unit disk 𝔻 of the complex plane in a recursive and linear way. We look for interpolating sequences ( z n ) in 𝔻 so that given a bounded sequence ( a n ) and a suitable sequence ( w n ), there is a bounded analytic function f on 𝔻 such that f ( z 1 ) = w 1 and f ( z n +1 ) = a n f ( z n ) + w n +1 . We add a recursion for the derivative of the type: f ′( z 1 ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msubsup> <m:mi>w</m:mi> <m:mn>1</m:mn> <m:mo>′</m:mo> </m:msubsup> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} w_1' \end{array} $ and f ′( z n +1 ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msubsup> <m:mi>a</m:mi> <m:mi>n</m:mi> <m:mo>′</m:mo> </m:msubsup> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} a_n' \end{array} $ [(1 − | z n | 2 )/(1 − | z n +1 | 2 )] f ′( z n ) + <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msubsup> <m:mi>w</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>′</m:mo> </m:msubsup> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} w_{n+1}', \end{array} $ where ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msubsup> <m:mi>a</m:mi> <m:mi>n</m:mi> <m:mo>′</m:mo> </m:msubsup> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} a_n' \end{array} $ ) is bounded and ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:msubsup> <m:mi>w</m:mi> <m:mi>n</m:mi> <m:mo>′</m:mo> </m:msubsup> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} w_n' \end{array} $ ) is an appropriate sequence, and we also look for zero-sequences verifying the recursion for f ′. The conditions on these interpolating sequences involve the Blaschke product with zeros at their points, one of them being the uniform separation condition.

Cited by 2Open access

Thin Interpolating Sequences

Pamela Gorkin, Brett D. Wick · Contemporary mathematics - American Mathematical Society · 2016

We look at thin interpolating sequences and the role they play in uniform algebras, Hardy spaces, and model spaces.

Cited by 1Open access

Interpolating sequences on curves

S. A. Obaid, D. C. Rung · Rocky Mountain Journal of Mathematics · 1985

We establish a condition on boundary curves (ending at points) lying either in the unit disc or the upper half plane which implies that any consecutively separated sequence, in the hyperbolic distance, lying on one of these curves is an interpolating sequence for bounded holomorphic functions.

Cited by 1Open access

Interpolating sequences in polydisks

Eric P. Kronstadt · Transactions of the American Mathematical Society · 1974

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{D^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the unit polydisk in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {C}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a uniform algebra, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity Baseline left-parenthesis upper D Superscript n Baseline comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }({D^n},A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the space of bounded analytic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> valued functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{D^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Superscript normal infinity Baseline upper A"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{l^\infty }A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the space of bounded sequences of elements in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A sequence, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals left-brace a Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity Baseline subset-of upper D Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will be called an interpolating sequence with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper H Superscript normal infinity Baseline left-parenthesis upper D Superscript n Baseline comma upper A right-parenthesis right-arrow l Superscript normal infinity Baseline upper A"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T:{H^\infty }({D^n},A) \to {l^\infty }A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis f right-parenthesis equals left-brace f left-parenthesis a Subscript i Baseline right-parenthesis right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">T(f) = \{ f({a_i})\} _{i = 1}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is surjective. In 1958, L. Carleson showed that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 1 comma upper S"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n = 1,S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is interpolating w.r.t. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {C}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> iff <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A not-equals bold upper C"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>≠<!-- ≠ --><

Cited by 4Open access

Interpolating Sequences in Polydisks

Eric P. Kronstadt · Transactions of the American Mathematical Society · 1974

Let Dn be the unit polydisk in c", A be a uniform algebra

Cited by 0

Interpolating and sampling sequences for . . .

Nicolas Marco, Xavier Massaneda, Joaquim Ortega-cerdà · 2002

We characterise interpolating and sampling sequences for the spaces of entire functions f such that fe −φ ∈ L p (C), p ≥ 1 (and some related weighted classes), where φ is a subharmonic weight whose Laplacian is a doubling measure. The results are expressed in terms of some densities adapted to the metric induced by ∆φ. They generalise previous results by Seip for the case φ(z) = |z | 2, and by Berndtsson &amp;amp; Ortega-Cerdà and Ortega-Cerdà &amp;amp; Seip for the

Cited by 0

Interpolating sequences and extremal problems

Rare & Special e-Zone (The Hong Kong University of Science and Technology) · 2014

b1334484 HKUST Electronic Theses Interpolating sequences and extremal problems by Ching-Cheong Lee thesis 2014 vii, 48 pages : illustrations ; 30 cm In 1958 Carleson proved that a sequence is interpolating if it is uniformly separated, later in 1983 Peter…Read more ›

Cited by 0

Sequences interpolating some geometric inequalities

Dorin Andrica, Ştefan Marinescu · Creative Mathematics and Informatics · 2019

Using the geometric dynamic of an iterative process (Theorem 2.1), we obtain refinements to some famous geometric inequalities in a triangle by constructing interpolating sequences.

Cited by 2Open access

Interpolating Sequences in the Polydisc

Bo Berndtsson, Sun‐Yung A. Chang, Kai-Ching Lin · Transactions of the American Mathematical Society · 1987

Let H(Dn) denote the set of all bounded analytic functions defined on the polydisc D" of Cn.In this note, we give a sufficient condition for sequences of points in >" to be interpolating sequences for H(Dn).We also discuss some conditions for interpolation of general domains.

Cited by 7Open access

INTERPOLATING SEQUENCES ON THE BIDISK

Jim Agler, John E. McCarthy · International Journal of Mathematics · 2001

We give a characterization of interpolating sequences for bounded analytic functions on the bidisk.

Cited by 0Open access

Interpolating sequences in the polydisc

Bo Berndtsson, Sun‐Yung A. Chang, Kai-Ching Lin · Transactions of the American Mathematical Society · 1987

Let H°°(Dn) denote the set of all bounded analytic functions defined on the polydisc D" of Cn.In this note, we give a sufficient condition for sequences of points in £>" to be interpolating sequences for H°°(Dn).We also discuss some conditions for interpolation of general domains.

Cited by 35