Recursive interpolating sequences

Francesc Tugores

doi:10.1515/math-2018-0044

Article Open Access

Recursive interpolating sequences

Francesc Tugores

Published/Copyright: April 30, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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(z₁) = w₁ and f(z_n+1) = a_nf(z_n) + w_n+1. We add a recursion for the derivative of the type: f′(z₁) = w1′ and f′(z_n+1) = an′ [(1 − |z_n|²)/(1 − |z_n+1|²)] f′(z_n) + wn+1′, where (an′) is bounded and (wn′) 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.

Keywords: Interpolating sequence; Uniformly separated sequence; Bounded analytic function

MSC 2010: 30E05; 30H05; 30J10

1 Introduction

Interpolation problems on the unit disk 𝔻 are a classical branch of complex analysis. Several types of interpolating sequences for different classes of analytic functions have been addressed since the middle of the last century, beginning with the celebrated works of W.K. Hayman [1], D.J. Newman [2] and L. Carleson [3] about the so-called “universal” interpolation problem, which consists in characterizing the sequences (z_n) in 𝔻 verifying that for any bounded sequence (w_n), there is a bounded analytic function f on 𝔻 such that f(z_n) = w_n.

On the other hand, recursion appears in many areas of mathematics: formulas, algorithms, optimization…, providing alternative definition procedures. Since there exists a specific theory for recursive numerical sequences and interpolating sequences have not been studied from a recursive perspective, we think it is interesting to pose recursive-type interpolation problems.

We want to emphasize that our approach converts the universal interpolation problem and other problems related to it into trivial cases of those that we introduce. Furthermore, most conditions involved are new and depend not only on the separation of the points of the sequence in 𝔻, but also on the sequences that we employ to define recursion.

We begin with the necessary notation. Let H^∞ be the space of all analytic functions f on 𝔻 such that ‖f‖_∞ = sup_z∈𝔻|f(z)| < ∞ and let l^∞ be the Banach space of all sequences of complex numbers (w_n) such that ‖(w_n)‖_∞ = sup_n |w_n| < ∞. We put Z = (z_n) for any sequence of different points in 𝔻 verifying the Blaschke condition ∑_n(1 − |z_n|) < ∞, which characterizes the zero-sequences of functions in H^∞. For two points z and w in 𝔻, we write

ψ(z,w)=z−w1−z¯w,

so that ρ = |ψ| is their pseudo-hyperbolic distance. Let B be the Blaschke product with zeros at Z, that is,

B(z)=∏n|zn|znψ(zn,z).

If E is a subsequence of Z, we put B_E for the Blaschke product with zeros at E and for a fixed m ∈ ℕ, we denote B_{Z∖{z_m}} by B_m. We write c for strictly positive constants that may change from one occurrence to the next one.

First, we recall that Z is interpolating if given any (w_n) ∈ l^∞, there exists f ∈ H^∞ such that f(z_n) = w_n. Interpolating sequences are characterized by the well-known Carleson’s theorem:

Theorem 1.1

([3]). Zis interpolating if and only if

|Bm(zm)|≥c∀m∈N.(1)

Sequences satisfying (1) are called uniformly separated (u.s.). From the Schwarz lemma, it follows that

|f(z)−f(w)|≤c‖f‖∞ρ(z,w)(2)

and thus, it is said that Z is interpolating in differences if given (w_n) verifying |w_i − w_j| ≤ cρ(z_i, z_j), there is f ∈ H^∞ such that f(z_n) = w_n. These sequences are the union of two u.s. [4], characterized as follows.

Lemma 1.2

([5]).

For a sequenceZ, the following are equivalent

Zis the union of two u.s. sequences.
For eachz_i, there existsz_jsuch that |B_i(z_i)| ≥ cρ(z_i, z_j).
Zis either u.s. or it can be rearrangedZ = (α_n) ∪ (β_n), where (α_n) and (β_n) are u.s. sequences, ρ(α_n,β_n) = ρ(α_n,Z ∖ {α_n}) → 0, andρ(α_n, z_i) ≥ c, ifz_i ≠ β_n.

Finally, since |f′(z)|(1 − |z|²) ≤ c ‖f‖_∞, it is said that Z is double interpolating if given (w_n) ∈ l^∞ and (wn′) satisfying (wn′(1 − |z_n|²)) ∈ l^∞, there is f ∈ H^∞ such that f(z_n) = w_n and f′(z_n) = wn′. It is proved in [6] that these sequences are also the u.s. ones.

Next we consider the following three quantities for the terms of a sequence T = (t_n) ∈ l^∞:

Γ(tm)=ρ(zm,zm+1)+|1−tm|Π(tm)=‖T‖∞ρ(zm,zm+1)+ρ(zm+1,zm+2)+|tm−tm+1|Λ(tm)=ρ(zm,zm+1){ρ(zm,zm+1)+|1+tm|}.

We need them to take suitable target spaces in our interpolation problems and they also appear in the results we get (Section 2).

We write Γ(t_m) = O(Γ(t_m+1)) (resp. Λ(t_m) = O(Λ(t_m+1))) if there exists a constant c_{T, Z} > 0 such that Γ(t_m) ≤ c_{T, Z}Γ(t_m+1) (resp. Λ(t_m) ≤ c_{T, Z}Λ(t_m+1)) for all m ∈ ℕ. We write Π(t_m) ∼ Π(t_m+1) if Π(t_m) ≤ c_{T, Z}Π(t_m+1) and Π(t_m+1) ≤ cT,Z′Π(t_m) for some constants c_{T, Z}, cT,Z′ > 0 and for all m ∈ ℕ.

From now on A = (a_n) and A′ = (an′) will denote sequences in l^∞. Our purpose is to examine the following distinguished sequences of 𝔻:

Definition 1.3

We say thatZisA-interpolating if given (w_n) verifying

|(∑i=1naiai+1…anwi)+wn+1|≤c(3)

and

|wn+1|≤cΓ(an),(4)

there existsf ∈ H^∞such that

{f(z1)=w1f(zn+1)=anf(zn)+wn+1.(5)

Clearly if ‖A‖_∞ < 1, then the sum in (3) is bounded by ‖(wn)‖∞1−‖A‖∞.

Definition 1.4

We say thatZ is A-interpolating in differences if given (w_n) satisfying(3)and

|wn+1−wn+2|≤cΠ(an),(6)

there isf ∈ H^∞verifying recursion(5).

Definition 1.5

We say thatZis (A,A′)-interpolating if given (w_n) satisfying(3)and(4)and (wn′) verifying

|∑i=1nai′ai+1′…an′(1−|zi|2)wi′+(1−|zn+1|2)wn+1′|≤c(7)

and

|wn+1′|(1−|zn+1|2)≤cΓ(an′),(8)

there existsf ∈ H^∞satisfying recursion(5)and

{f′(z1)=w1′f′(zn+1)=an′1−|zn|21−|zn+1|2f′(zn)+wn+1′.(9)

Definition 1.6

We say thatZis zero andA′-interpolating if given (wn′) verifying(7),

|w1′|(1−|z1|2)≤cΛ(a1′)(10)

and

|wn+1′|(1−|zn+1|2)≤cΛ(an′),(11)

there isf ∈ H^∞vanishing onZand satisfying recursion(9).

Recursion (5) is equivalent to f(z_n) = μ_n, where

{μ1=w1μn+1=(∑i=1naiai+1…anwi)+wn+1,(12)

and recursion (9) is equivalent to f′(z_n) = μn′, with

{μ1′=w1′μn+1′=∑i=1nai′ai+1′…an′(1−|zi|2)wi′1−|zn+1|2+wn+1′.(13)

Thus, we must have (3) and (7) to state that sequences (μ_n) and (μn′(1 − |z_n|²)) are bounded. We impose that data sequences (w_n) and (wn′) verify (4), (6), (8) and (11), because they are intrinsic to recursions (a technical reason justifies (10)). In effect, (4) and (6) are obtained taking into account (2) in the inequalities

|wn+1|≤|f(zn)−f(zn+1)|+|f(zn)||1−an|

and

|wn+1−wn+2|≤|an||f(zn)−f(zn+1)|+|f(zn+1)−f(zn+2)|+|f(zn+1)||an−an+1|,

respectively. On the other hand,

|f′(z)(1−|z|2)−f′(w)(1−|w|2)|≤c‖f‖∞ρ(z,w),(14)

|f(z)−f(w)+f′(w)(1−|w|2)ψ(z,w)|≤c‖f‖∞ρ(z,w)2(15)

and

|f(z)−f(w)−[f′(z)(1−|z|2)+f′(w)(1−|w|2)]ψ(z,w)2|≤c‖f‖∞ρ(z,w)3.(16)

See [7] for (14), [8] for (15) and [9] for (16). Thus, (8) is obtained using (14) in the inequality

|wn+1′|(1−|zn+1|2)≤|f′(zn+1)(1−|zn+1|2)−f′(zn)(1−|zn|2)|+|f′(zn)|(1−|zn|2)|1−an′|.

If on the right of this last inequality, we put

|f′(zn+1)(1−|zn+1|2)+f′(zn)(1−|zn|2)|+|f′(zn)|(1−|zn|2)|1+an′|,

then (11) is obtained using (16) for the first summand and (15) for the second one.

We introduce these interpolating sequences because they provide a generalization of the usual interpolation problems, in the sense that (0)-interpolating sequences, (0)-interpolating in differences and ((0), (0))-interpolating are interpolating, interpolating in differences and double interpolating, respectively.

Extending recursion to an arbitrary order or increasing the degree of derivability are projects certainly cumbersome, so that we confine ourselves to order one and the first derivative. Nevertheless, we think it would be interesting to consider these types of sequences for other spaces of analytic functions, such as the Lipschitz class and the Bloch space, for which the pseudo-hyperbolic distance in (2) is replaced by the Euclidean and hyperbolic distance, respectively (interpolating sequences for these spaces are characterized in [10] and [11]).

While the proofs of results turn out to be rather standard (Carleson’s theorem is used repeatedly), we appreciate the following separation conditions, which are consistent with the problems posed and appear in a natural way.

Definition 1.7

We say that (Z, A) satisfies condition (S) if

|Bm+1(zm+1)|≥cΓ(am)∀m∈N,

and condition (D) if

|Bm+1(zm+1)|≥cΠ(am)∀m∈N.

We say that (Z, A′) satisfies condition (M) if

|Bm+1(zm+1)|2≥cΛ(am′)∀m∈N.

We name (S), (D) and (M) to the above conditions because these are the initials of simple, differences and mixed, respectively, and we will see in the next section that condition (S) is related to interpolating sequences in a simple sense; (D), in a differences sense, and (M), in a mixed sense (zero and interpolating).

Since ρ(z_m, z_m+1) > |B_m+1(z_m+1)|, it follows that if (Z, A′) verifies (M), then (Z, −A′) satisfies (S). All conditions imply (b) in Lemma 1.2 so that Z is the union of two u.s. sequences. Note that if ‖A‖_∞ < 1 (resp. ‖ A′‖_∞ < 1), then (S) (resp. (M)) ⇒ u.s.

2 Statement of results

Our results are the following ones.

Proposition 2.1

IfZisA-interpolating andΓ(a_n) = O(Γ(a_n+1)), then (Z, A) verifies (S).
IfZisA-interpolating in differences andΠ(a_n) ∼ Π(a_n+1), then (Z, A) verifies (D).
IfZis (A, A′)-interpolating, Γ(a_n) = O(Γ(a_n+1)) andΓ(an′)=O(Γ(an+1′)),thenZis u.s.
IfZis zero andA′-interpolating andΛ(an′)=O(Λ(an+1′)),then (Z, A′) verifies (M).

Proposition 2.2

If (Z, A) verifies (S), thenZisA-interpolating.
If (Z, A) verifies (D) andAis such that
|an+1|≤r|an|(17)
for some r ∈ (0, 1), thenZisA-interpolating in differences.
IfZis u.s., thenZis (A, A′)-interpolating for any sequencesAandA′.
If (Z, A′) verifies (M), Λ(an′)=O(Λ(an+1′))and
|a1′a2′⋯an′|Λ(a1′)+∑i=2n|ai′ai+1′⋯an′|Λ(ai−1′)≤cΛ(an−1′)∀n≥2,(18)
thenZis zero andA′-interpolating.

3 Proof of results

Proof of proposition 2.1

For a fixed m ∈ ℕ, let (w_n) be defined by w_m+1 = Γ(a_m), w_m+2 = −a_m+1w_m+1 and w_n = 0 if n ≠ m + 1, m + 2. Since Γ(a_n) = O(Γ(a_n+1)), it follows that
|wm+2|≤‖A‖∞Γ(am)≤cA,Z‖A‖∞Γ(am+1)
and w_m+2 also verifies (4). Since the operator
R:H∞⟶{(wn)/(wn) verifies (3) and (4)}
defined by 𝓡(f) = (u_n), where u₁ = f(z₁) and u_n+1 = f(z_n+1) −a_nf(z_n), is linear and onto, a standard argument using the closed graph theorem gives the existence of f_m ∈ H^∞ such that 𝓡(f_m) = (w_n) and ‖ f_m‖_∞ ≤ c ‖ (w_n)‖_∞ = c. Since f_m(z_m+1) = w_m+1 and f_m(z_n) = 0 if n ≠ m + 1, there is g_m ∈ H^∞ such that f_m = g_mB_m+1 and ‖ g_m‖_∞ = ‖ f_m‖_∞. Thus,
Γ(am)=|fm(zm+1)|≤c|Bm+1(zm+1)|
and (S) holds.
For a fixed m ∈ ℕ, let (w_n) be defined by w_m+1 = Π(a_m) and the other terms as in (i). This sequence satisfies (6), because taking into account that Π(a_n) ∼ Π(a_n+1),
|wm+1−wm+2|=|1+am+1|wm+1≤(1+‖A‖∞)Π(am),|wm−wm+1|=wm+1=Π(am)≤cA,Z′Π(am−1) if m≥2,|wm+2−wm+3|=wm+2≤‖A‖∞Π(am)≤cA,Z‖A‖∞Π(am+1).
Condition (D) is obtained proceeding exactly as in the proof of (i).
Let (w_n) be defined as in (i) and let (wn′) be defined by wm+1′=Γ(am′)1−|zm+1|2,wm+2′=−am+1′Γ(am′)1−|zm+2|2 and wn′ = 0 if n ≠ m + 1, m + 2. We have that wm+2′ also satisfies (8), because
|wm+2′|(1−|zm+2|2)≤‖A′‖∞Γ(am′)≤cA′,Z‖A′‖∞Γ(am+1′).
Proceeding as in (i), there exists g_m ∈ H^∞ such that fm=gmBm+12. Since ρ(z_m, z_m+1) ≤ Γ(a_m) (see definition of Γ) and |B_m+1(z_m+1)| < ρ(z_m, z_m+1), we have
ρ(zm,zm+1)≤Γ(am)=|fm(zm+1)|≤c|Bm+1(zm+1)|2<c|Bm+1(zm+1)|ρ(zm,zm+1).
Thus, it follows that Z ∖ {z₁} is u.s. and so is Z.
Let (wn′) be defined as in (iii) replacing Γ by Λ. Since Λ(an′)=O(Λ(an+1′)),thenwm+2′ verifies (11). Proceeding as in (i), there is g_m ∈ H^∞ such that f_m = g_mBB_m+1. A simple calculation gives
|B′(zm+1)|=|Bm+1(zm+1)|1−|zm+1|2(19)
and then,
Λ(am′)1−|zm+1|2=|fm′(zm+1)|=|(gmB′Bm+1)(zm+1)|≤c|Bm+1(zm+1)|21−|zm+1|2.
Thus, (M) holds.

□

Proof of proposition 2.2

Let (μ_n) be as in (12) and (μn′) as in (13).

We situate ourselves in (c) of Lemma 1.2. If Z is u.s., then Carleson’s theorem provides f performing the interpolation f(z_n) = μ_n. Otherwise, Z is the union of E = (z_2m) and F = (z_2m+1), both u.s. Let g ∈ H^∞ such that g(z_2m) = μ_2m. We look for a function h ∈ H^∞ such that h(z_2m+1) = λ_m, where
λm=μ2m+1−g(z2m+1)BE(z2m+1),
because then f₁ = g + B_Eh is in H^∞ and performs the above interpolation on Z ∖ {z₁}. Since |B_E(z_2m+1)| > |B_2m+1(z_2m+1)| and |B_2m+1(z_2m+1)| ≥ cΓ(a_2m) (condition (S)), we have |B_E(z_2m+1)| ≥ cΓ(a_2m). Then,
|λm|≤c|μ2m+1−μ2m|+|g(z2m)−g(z2m+1)|Γ(a2m).
Taking into account that
μ2m+1=a2mμ2m+w2m+1(20)
and using (3) and (4),
|μ2m+1−μ2m|≤|1−a2m||μ2m|+|w2m+1|≤cΓ(a2m).
By (2)
|g(z2m)−g(z2m+1)|≤cρ(z2m,z2m+1)≤cΓ(a2m).
Thus, (λ_n) ∈ l^∞ and Carleson’s theorem provides h. Putting
f(z)=f1(z)+(μ1−f1(z1))B1(z)B1(z1),
it follows that f is in H^∞ and performs the above interpolation on Z.
The proof is as in (i). Now we have
|λm|≤c|μ2m+1−μ2m+2|+|g(z2m+2)−g(z2m+1)|Π(a2m).
By (20) and using (3) and (6),
|μ2m+1−μ2m+2|≤|w2m+1−w2m+2|+|μ2m+1||a2m−a2m+1|+|a2m|(|μ2m+1|+|μ2m|)≤c{Π(a2m)+|a2m|}
and by (17), it turns out that |a2m|≤|a2m−a2m+1|1−r≤cΠ(a2m). On the other hand, by (2)
|g(z2m+2)−g(z2m+1)|≤cρ(z2m+1,z2m+2)≤cΠ(a2m).
Thus, (λ_n) ∈ l^∞ and the proof continues as in (i).
Since Z is u.s., then Z is double interpolating and there is f in H^∞ performing f(z_n) = μ_n and f′(z_n) = μn′.
In case that Z = E ∪ F, where E = (z_2m) and F = (z_2m+1) are u.s., we know that there is g ∈ H^∞ vanishing on E and satisfying g′(z_2m) = μ2m′. We look for a function h₁ ∈ H^∞ such that h₁(z_2m+1) = λ_m, where
λm=−g(z2m+1)BE2(z2m+1),
because then h2=g+BE2h1 is in H^∞, vanishes on Z ∖ {z₁} and h2′(z2m)=μ2m′. We have
|λm|≤c|g(z2m+1)|Λ(a2m′).
It follows from (13), taking into account (10) and (11), that
(1−|z2|2)|μ2′|≤|a1′|(1−|z1|2)|w1′|+(1−|z2|2)|w2′|≤cΛ(a1′).(21)
For n ≥ 2, by (13), (10), (11) and (18)
(1−|zn+1|2)|μn+1′|≤c{|a1′a2′…an′|Λ(a1′)+∑i=2n|ai′ai+1′…an′|Λ(ai−1′)}+(1−|zn+1|2)|wn+1′|≤c{Λ(an−1′)+Λ(an′)}≤cΛ(an′).(22)
Then, using (15), (21), (22) and Λ(an′)=O(Λ(an+1′)),
|g(z2m+1)|=|g(z2m)−g(z2m+1)−g′(z2m)(1−|z2m|2)ψ(z2m,z2m+1)|+(1−|z2m|2)|μ2m′|ρ(z2m,z2m+1)≤c{ρ(z2m,z2m+1)2+Λ(a2m′)ρ(z2m,z2m+1)}.
Thus,
|λm|≤c(ρ(z2m,z2m+1)2Λ(a2m′)+ρ(z2m,z2m+1))≤c
and Carleson’s theorem provides h₁ performing the above interpolation. We replace h₂ by h_2c, where h_2c is given by
h2c(z)=h2(z)−h2(z1)B12(z)B12(z1),
because then h_2c also vanishes on z₁. Finally, we look for h₃ ∈ H^∞ such that h₃(z_2m+1) = λm′, where
λm′=μ2m+1′−h2c′(z2m+1)(B′BE)(z2m+1),
because then f = h_2c + BB_Eh₃ is in H^∞ and performs the desired interpolation on Z ∖ {z₁}. Using (19),
|λm′|≤c{|μ2m+1′+μ2m′|+|h2c′(z2m)+h2c′(z2m+1)|}(1−|z2m+1|2)Λ(a2m′).
An easy computation shows that
|z−w|1−|z|2<4ρ(z,w),(23)
if ρ(z, w) < min(|z|,12); then, using (21), (22), (23) and Λ(an′)=O(Λ(an+1′)),
|μ2m′|||z2m|2−|z2m+1|2|≤cΛ(a2m−1′)|z2m−z2m+1|1−|z2m|2≤cΛ(a2m′)ρ(z2m,z2m+1)(24)
and taking into account (21), (22), (24) and Λ(an′)=O(Λ(an+1′)),
|μ2m+1′+μ2m′|(1−|z2m+1|2)≤|μ2m+1′|(1−|z2m+1|2)+|μ2m′|(1−|z2m|2)+|μ2m′|||z2m|2−|z2m+1|2|≤c{Λ(a2m′)+Λ(a2m′)ρ(z2m,z2m+1)}.
On the other hand, by (16) and (24)
|h2c′(z2m)+h2c′(z2m+1)|(1−|z2m+1|2)≤|h2c′(z2m)(1−|z2m|2)+h2c′(z2m+1)(1−|z2m+1|2)|+|μ2m′|||z2m|2−|z2m+1|2|≤c{ρ(z2m,z2m+1)2+Λ(a2m′)ρ(z2m,z2m+1)}.
Then,
|λm′|≤c(1+ρ(z2m,z2m+1)2Λ(a2m′)+ρ(z2m,z2m+1))≤c
and Carleson’s theorem provides h₃ performing the above interpolation. We replace f by f_c defined by
fc(z)=f(z)−(f′(z1)−μ1′)(BB1)(z)(B′B1)(z1),
because then fc′(z1)=μ1′ and so f_c performs the desired interpolation on Z.

□

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Received: 2017-02-28

Accepted: 2018-03-28

Published Online: 2018-04-30

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0044

Keywords for this article

Interpolating sequence; Uniformly separated sequence; Bounded analytic function

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BY-NC-ND 4.0