On Interpolation by Functions in p A Raymond Cheng and Christopher Felder

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On Interpolation by Functions in `p

Raymond Cheng and Christopher Felder

Abstract. This work explores several aspects of interpolating sequences for `p

A, the space

of analytic functions on the unit disk with p-summable Maclaurin coeﬃcients. Much of

this work is communicated through a Carlesonian lens. We investigate various analogues of

Gramian matrices, for which we show boundedness conditions are necessary and suﬃcient for

interpolation, including a characterization of universal interpolating sequences in terms of

Riesz systems. We also discuss weak separation, giving a characterization of such sequences

using a generalization of the pseudohyperbolic metric. Lastly, we consider Carleson measures

and embeddings.

1. Introduction

Let {zk}∞

k=0 be a sequence of distinct points in the open unit disk Dof the complex plane, and

let {wk}∞

k=0 be a sequence of complex numbers. An interpolation problem, broadly speaking,

is to ﬁnd an analytic function fon Dsuch that

f(zk) = wk

for all k= 0,1,2, . . .. Furthermore, it is natural to seek a characterization of those sequences

{zk}and {wk}for which such a function falways exists, and belongs to a certain class of

analytic functions on D. As explained by Duren [15, p. 148], intuition suggests that the

points {zk}must not lie too “close together,” lest a highly oscillatory choice of targets {wk}

fail to be interpolated by a function of the prescribed class.

This notion is illustrated by a theorem of Carleson [5], which characterizes interpolation by

functions in H∞. A sequence {zk}in Dis said to be uniformly separated if there exists δ > 0

such that ∞

j=0, j6=k

zk−zj

1−¯zjzk>δ

for all k= 1,2,3, . . ..

Theorem 1.1 (Carleson).Let {zk}be a sequence of points in D. Then {zk}has the property

that for any bounded sequence {wk}of complex numbers, there exists f∈H∞such that

f(zk) = wk

2020 Mathematics Subject Classiﬁcation. Primary 46E15; Secondary 30J99, 30H99.

arXiv:2210.05823v1 [math.FA] 11 Oct 2022

for all k= 1,2,3, . . ., if and only if {zk}is uniformly separated.

This was extended to Hpby Shapiro and Shields [22]. See the books by Seip [21], and by

Agler and McCarthy [1], for a modern exposition of interpolation in numerous other spaces

of functions.

The present paper is concerned with interpolation by functions in `p

A, the space of analytic

functions in Dwhose Maclaurin coeﬃcients are p-summable, with 1 < p < ∞. Vinogradov

[23, 24] (for English translations, with Khavin, see [25, 26]) derived exact conditions for

interpolation in `p

Aprovided that the sequence {zk}lies in a Stoltz domain, or that it tends

rapidly enough to the boundary. Our approach connects interpolation in `p

Awith associated

sequences of functionals, inﬁnite matrices, a nonlinear functional equation, notions of sepa-

ration, and Carleson measures. In particular, after providing some background information

on the `p

Aspaces in the next section, we move to Section 3, which is concerned with various

upper and lower bounds that characterize universal interpolating sequences. Section 4 in-

volves studying limits of truncated interpolation problems to deduce an interpolation result

based on the geometry of the Banach space in terms of a minimality condition. Criteria for

interpolation are expressed in terms of matrix conditions in Section 5. There arises a pair of

nonlinear operators that extend the notion of a Gramian matrix to the case p6= 2. Section

6 characterizes sequences which are weakly separated by the multiplier algebra of `p

A. We

close with a section on Carleson measures for `p

A, and a handful of open questions.

2. Preliminaries

For 0 < p 6∞, the space `p

Ais deﬁned to be space of analytic functions on the open unit

disk Dfor which the Maclaurin coeﬃcients are pth order summable, i.e.,

A:= (f(z) = X

n≥0

anzn∈Hol(D) : X

n≥0

|an|p<∞).

This function space is endowed with the norm (or quasinorm, if 0 < p 61) that it inherits

from the sequence space `p. Thus let us write

kfkp=k{ak}∞

k=0k`p

for any f(z) = P∞

k=0 akzkbelonging to `p

A. When p= 2, we recover the classical Hardy

space H2on the disk, however, we emphasize that k·kprefers to the norm on `p

A, and not

the norm on the Hardy space Hp, or some other function space parametrized by p.

We limit our attention to the range 1 < p < ∞. For then `p

Ais reﬂexive, smooth and

uniformly convex. In particular, it enjoys the unique nearest point property, and each

nonzero vector has a unique norming functional.

Throughout this paper, qwill be the H¨older conjugate to p, that is, 1/p + 1/q = 1 holds.

We recall that for 1 6p < ∞,p6= 2, the dual space of `p

Acan be identiﬁed with `q

A, under

the pairing

hf, gi=

∞

k=0

fkgk,

where f(z) = P∞

k=0 fkzk∈`p

Aand g(z) = P∞

k=0 gkzk∈`q

Point evaluation at any w∈Dis a bounded linear functional on `p

A. It is implemented by

the kernel function Λw∈`q

Agiven by

Λw(z) :=

∞

k=0

wkzk, z ∈D.

Indeed, for any f(z) = P∞

k=0 akzk∈`p

Awe have

f(w) = hf, Λwi=

∞

k=0

akwk.

The norm of Λw(as either a vector or a functional) is

kΛwkq=1

(1 − |w|q)1/q .

There is a sensible way to deﬁne “inner function” in the context of `p

A, that employs a notion

of orthogonality in general normed linear spaces.

Let xand ybe vectors belonging to a normed linear space X. We say that xis orthogonal

to yin the Birkhoﬀ-James sense [3, 18] if

(2.1) kx+βykX>kxkX

for all scalars β, and in this case we write x⊥Xy. Birkhoﬀ-James orthogonality is one way

to extend the concept of orthogonality from an inner product space to normed spaces. There

are other ways to generalize orthogonality, but this approach is particularly useful since it is

relates directly to an extremal condition, namely (2.1).

If Xis a Hilbert space, then the usual orthogonality relation x⊥yis equivalent to x⊥Xy.

More generally, however, the relation ⊥Xis neither symmetric nor linear. When X=`p

A, let

us write ⊥pinstead of ⊥`p

A. There is a practical criterion for the relation ⊥pwhen 1 <p<∞.

Theorem 2.2 (James [18]).Suppose that 1< p < ∞. Then for f(z) = P∞

k=0 fkzkand

g(z) = P∞

k=0 gkzkbelonging to `p

Awe have

(2.3) f⊥pg⇐⇒

∞

k=0

|fk|p−2fkgk= 0,

where any occurrence of “|0|p−20” in the right side is interpreted as zero.

In light of (2.3) we deﬁne, for a complex number α=reiθ, and any s > 0, the quantity

(2.4) αhsi= (reiθ)hsi:= rse−iθ.

It is readily seen that for any complex numbers αand β, exponent s > 0, and integer n>0,

we have

(αβ)hsi=αhsiβhsi

|αhsi|=|α|s

αhsiα=|α|s+1

(αhsi)n= (αn)hsi

(αhp−1i)hq−1i=α.

Further to the notation (2.4), for f(z) = P∞

k=0 fkzk, let us write

(2.5) fhsi(z) :=

∞

k=0

fhsi

kzk

for any s > 0. Similarly, for any matrix or vector with complex-valued entries A= [aj,k],

we take Ahsito mean the matrix [ahsi

j,k]. By comparing with the case p= 2, we can think of

taking the hsipower as generalizing complex conjugation.

If f∈`p

A, it is easy to verify that fhp−1i∈`q

A. Thus from (2.3) we get

(2.6) f⊥pg⇐⇒ hg, fhp−1ii= 0.

Consequently the relation ⊥pis linear in its second argument, when 1 <p<∞, and it

then makes sense to speak of a vector being orthogonal to a subspace of `p

A. In particular, if

f⊥pgfor all gbelonging to a subspace Mof `p

A, then

kf+gkp>kfkp

for all g∈M. That is, fsolves a nearest-point problem in relation to the subspace M.

Direct calculation will also conﬁrm that

hf, fhp−1ii=kfkp

and hence fhp−1i/kfkp−1

p∈`q

Ais the norming functional of f∈`p

A\ {0}(smoothness ensures

that the norming functional is unique). With this concept of orthogonality established, we

may now introduce a deﬁnition of inner function that is particular to `p

Definition 2.7.Let 1 <p<∞. A function f∈`p

Ais said to be p-inner if it is not

identically zero and it satisﬁes

f(z)⊥pzkf(z)

for all positive integers k.

That is, fis p-inner if it is nontrivially orthogonal to all of its forward shifts. Apart from a

multiplicative constant, this deﬁnition, originating from [11], is equivalent to the traditional

meaning of “inner” when p= 2. Furthermore, this approach to deﬁning an inner property

is consistent with that taken in other function spaces [2, 4, 13, 14, 16, 17, 19, 20].

Birkhoﬀ-James Orthogonality also plays a role when we utilize a version of the Pythagorean

theorem for `p

A. It takes the form of a family of inequalities relating the lengths of orthogonal

vectors with that of their sum [8, Corollary 3.4].

Theorem 2.8.Let 1< r < ∞, and 1< p < ∞. If 1< p 626r < ∞or 26p6r < ∞,

then there exists K > 0such that

(2.9) kfkr

p+Kkgkr

p6kf+gkr

whenever f⊥pgin `p

If 1< r 6p62or 1< r 626p < ∞, then there exists K > 0such that

(2.10) kfkr

p+Kkgkr

p>kf+gkr

whenever x⊥pgin `p

Actually, this theorem holds with any Lebesgue space Lpin place of `p

A. When p= 2, the

parameters are K= 1 and r= 2, and the Pythagorean inequalities reduce to the familiar

Pythagorean theorem for a Hilbert space. More generally, these Pythagorean inequalities

enable the application of some Hilbert space methods and techniques to smooth Banach

spaces satisfying the weak parallelogram laws; see, for example, [12, Proposition 4.8.1 and

Proposition 4.8.3; Theorem 8.8.1].

For further background on the function space `p

A, we refer to the book [12].

3. Boundedness Conditions

Much of the modern treatment of interpolating sequences has been connected to the bound-

edness of Gramian matrices. For example, in H∞, Carleson’s theorem tells us that a sequence

{zk} ⊆ Dis interpolating if and only if the corresponding Gramian G, given by

Gi,j =hsi, sji2,

where siis the normalized Szeg˝o kernel at zi, is both bounded and bounded below (see [1,

Chapter 9] for a more general view). However, the boundedness of Gis equivalent to the

existence of a positive constant Msuch that



X

aisi



2≤MX

|ai|2,

for any sequence {ai}of scalars. When p6= 2, the conditions in the following propositions,

using H¨older duality, should be read as replacements for this phenomenon. We focus on a

direct matrix interpretation of this in Section 5.

Let us begin by interpolating a ﬁxed ﬁnite target when 1 < p < ∞,p6= 2, and letting

1/p + 1/q = 1. Fixing a positive integer N, distinct points z0, z1, z2, . . . , zN∈Dand points

w0, w1, w2, . . . , wN∈D, we should ﬁrst convince ourselves that there exists h∈`p

Asuch that

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OnInterpolationbyFunctionsin`pARaymondChengandChristopherFelderAbstract.Thisworkexploresseveralaspectsofinterpolatingsequencesfor`pA,thespaceofanalyticfunctionsontheunitdiskwithp-summableMaclaurincoecients.MuchofthisworkiscommunicatedthroughaCarlesonianlens.WeinvestigatevariousanaloguesofGramianmat...

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