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

$$\begin{array}{}{\displaystyle \psi (z,w)=\frac{z-w}{1-\overline{z}w},}\end{array}$$

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

$$\begin{array}{}{\displaystyle B(z)=\prod _{n}\frac{|{z}_{n}|}{{z}_{n}}\phantom{\rule{thinmathspace}{0ex}}\psi ({z}_{n},z).}\end{array}$$

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∖{zm}} 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]). *Z* *is interpolating if and only if*

$$\begin{array}{}{\displaystyle |{B}_{m}({z}_{m})|\ge c\phantom{\rule{1em}{0ex}}\mathrm{\forall}m\in \mathbb{N}.}\end{array}$$(1)

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

$$\begin{array}{}{\displaystyle |f(z)-f(w)|\le c\phantom{\rule{thinmathspace}{0ex}}\Vert f{\Vert}_{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\rho (z,w)}\end{array}$$(2)

and thus, it is said that *Z* is *interpolating in differences* if given (*w*_{n}) verifying |*w*_{i} − *w*_{j}| ≤ *c* *ρ*(*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 sequence* *Z*, *the following are equivalent*

*Z* *is the union of two u*.*s*. *sequences*.

*For each* *z*_{i}, *there exists* *z*_{j} *such that* |*B*_{i}(*z*_{i})| ≥ *c* *ρ*(*z*_{i}, *z*_{j}).

*Z* *is either u*.*s*. *or it can be rearranged* *Z* = (*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, *and* *ρ*(*α*_{n}, *z*_{i}) ≥ *c*, *if* *z*_{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 ($\begin{array}{}{w}_{n}^{\prime}\end{array}$) satisfying ($\begin{array}{}{w}_{n}^{\prime}\end{array}$(1 − |*z*_{n}|^{2})) ∈ *l*^{∞}, there is *f* ∈ *H*^{∞} such that *f*(*z*_{n}) = *w*_{n} and *f*′(*z*_{n}) = $\begin{array}{}{w}_{n}^{\prime}\end{array}$. 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*^{∞}:

$$\begin{array}{}{\displaystyle \mathit{\Gamma}({t}_{m})=\rho ({z}_{m},{z}_{m+1})+|1-{t}_{m}|}\\ \mathit{\Pi}({t}_{m})=\Vert T{\Vert}_{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\rho ({z}_{m},{z}_{m+1})+\rho ({z}_{m+1},{z}_{m+2})+|{t}_{m}-{t}_{m+1}|\\ \mathit{\Lambda}({t}_{m})=\rho ({z}_{m},{z}_{m+1})\phantom{\rule{thinmathspace}{0ex}}\{\rho ({z}_{m},{z}_{m+1})+|1+{t}_{m}|\}.\end{array}$$

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}) ≤ $\begin{array}{}{c}_{T,Z}^{\prime}\end{array}$ *Π*(*t*_{m}) for some constants *c*_{T, Z}, $\begin{array}{}{c}_{T,Z}^{\prime}\end{array}$ > 0 and for all *m* ∈ ℕ.

From now on *A* = (*a*_{n}) and *A*′ = ($\begin{array}{}{a}_{n}^{\prime}\end{array}$) will denote sequences in *l*^{∞}. Our purpose is to examine the following distinguished sequences of 𝔻:

#### Definition 1.3

*We say that* *Z* *is* *A*-*interpolating if given* (*w*_{n}) *verifying*

$$\begin{array}{}{\displaystyle |(\sum _{i=1}^{n}{a}_{i}\phantom{\rule{thinmathspace}{0ex}}{a}_{i+1}\dots {a}_{n}{w}_{i})+{w}_{n+1}|\le c}\end{array}$$(3)

*and*

$$\begin{array}{}{\displaystyle |{w}_{n+1}|\le c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Gamma}({a}_{n}),}\end{array}$$(4)

*there exists* *f* ∈ *H*^{∞} *such that*

$$\begin{array}{}{\displaystyle \{\begin{array}{l}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}f({z}_{1})={w}_{1}\\ \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}f({z}_{n+1})={a}_{n}f({z}_{n})+{w}_{n+1}.\end{array}}\end{array}$$(5)

Clearly if ‖*A*‖_{∞} < 1, then the sum in (3) is bounded by $\begin{array}{}{\displaystyle \frac{\Vert ({w}_{n}){\Vert}_{\mathrm{\infty}}}{1-\Vert A{\Vert}_{\mathrm{\infty}}}}.\end{array}$

#### Definition 1.4

*We say that* *Z is A*-*interpolating in differences if given* (*w*_{n}) *satisfying* (3) *and*

$$\begin{array}{}{\displaystyle |{w}_{n+1}-{w}_{n+2}|\le c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Pi}({a}_{n}),}\end{array}$$(6)

*there is* *f* ∈ *H*^{∞} *verifying recursion* (5).

#### Definition 1.5

*We say that* *Z* *is* (*A*,*A*′)-*interpolating if given* (*w*_{n}) *satisfying* (3) *and* (4) *and* ($\begin{array}{}{w}_{n}^{\prime}\end{array}$) *verifying*

$$\begin{array}{}{\displaystyle |\sum _{i=1}^{n}{a}_{i}^{\prime}\phantom{\rule{thinmathspace}{0ex}}{a}_{i+1}^{\prime}\dots {a}_{n}^{\prime}(1-|{z}_{i}{|}^{2}){w}_{i}^{\prime}+(1-|{z}_{n+1}{|}^{2}){w}_{n+1}^{\prime}|\le c}\end{array}$$(7)

*and*

$$\begin{array}{}{\displaystyle |{w}_{n+1}^{\prime}|(1-|{z}_{n+1}{|}^{2})\le c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Gamma}({a}_{n}^{\prime}),}\end{array}$$(8)

*there exists* *f* ∈ *H*^{∞} *satisfying recursion* (5) *and*

$$\begin{array}{}{\displaystyle \{\begin{array}{l}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{f}^{\prime}({z}_{1})={w}_{1}^{\prime}\\ \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{f}^{\prime}({z}_{n+1})={a}_{n}^{\prime}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{1-|{z}_{n}{|}^{2}}{1-|{z}_{n+1}{|}^{2}}}\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}({z}_{n})+{w}_{n+1}^{\prime}.\end{array}}\end{array}$$(9)

#### Definition 1.6

*We say that* *Z* *is zero and* *A*′-*interpolating if given* ($\begin{array}{}{w}_{n}^{\prime}\end{array}$) *verifying* (7),

$$\begin{array}{}{\displaystyle |{w}_{1}^{\prime}|(1-|{z}_{1}{|}^{2})\le c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Lambda}({a}_{1}^{\prime})}\end{array}$$(10)

*and*

$$\begin{array}{}{\displaystyle |{w}_{n+1}^{\prime}|(1-|{z}_{n+1}{|}^{2})\le c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Lambda}({a}_{n}^{\prime}),}\end{array}$$(11)

*there is* *f* ∈ *H*^{∞} *vanishing on* *Z* *and satisfying recursion* (9).

Recursion (5) is equivalent to *f*(*z*_{n}) = *μ*_{n}, where

$$\begin{array}{}{\displaystyle \{\begin{array}{l}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\mu}_{1}={w}_{1}\\ \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\mu}_{n+1}=(\sum _{i=1}^{n}{a}_{i}\phantom{\rule{thinmathspace}{0ex}}{a}_{i+1}\dots {a}_{n}{w}_{i})+{w}_{n+1},\end{array}}\end{array}$$(12)

and recursion (9) is equivalent to *f*′(*z*_{n}) = $\begin{array}{}{\mu}_{n}^{{}^{\prime}}\end{array}$, with

$$\begin{array}{}{\displaystyle \{\begin{array}{l}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\mu}_{1}^{\prime}={w}_{1}^{\prime}\\ \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\mu}_{n+1}^{\prime}={\displaystyle \frac{\sum _{i=1}^{n}{a}_{i}^{\prime}\phantom{\rule{thinmathspace}{0ex}}{a}_{i+1}^{\prime}\dots {a}_{n}^{\prime}(1-|{z}_{i}{|}^{2}){w}_{i}^{\prime}}{1-|{z}_{n+1}{|}^{2}}}+{w}_{n+1}^{\prime}.\end{array}}\end{array}$$(13)

Thus, we must have (3) and (7) to state that sequences (*μ*_{n}) and ($\begin{array}{}{\mu}_{n}^{{}^{\prime}}\end{array}$(1 − |*z*_{n}|^{2})) are bounded. We impose that data sequences (*w*_{n}) and ($\begin{array}{}{w}_{n}^{\prime}\end{array}$) 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

$$\begin{array}{}{\displaystyle |{w}_{n+1}|\le |f({z}_{n})-f({z}_{n+1})|+|f({z}_{n})|\phantom{\rule{thinmathspace}{0ex}}|1-{a}_{n}|}\end{array}$$

and

$$\begin{array}{}{\displaystyle |{w}_{n+1}-{w}_{n+2}|\le |{a}_{n}|\phantom{\rule{thinmathspace}{0ex}}|f({z}_{n})-f({z}_{n+1})|+|f({z}_{n+1})-f({z}_{n+2})|+|f({z}_{n+1})|\phantom{\rule{thinmathspace}{0ex}}|{a}_{n}-{a}_{n+1}|,}\end{array}$$

respectively. On the other hand,

$$\begin{array}{}{\displaystyle |{f}^{\prime}(z)(1-|z{|}^{2})-{f}^{\prime}(w)(1-|w{|}^{2})|\le c\phantom{\rule{thinmathspace}{0ex}}\Vert f{\Vert}_{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\rho (z,w),}\end{array}$$(14)

$$\begin{array}{}{\displaystyle |f(z)-f(w)+{f}^{\prime}(w)(1-|w{|}^{2})\psi (z,w)|\le c\phantom{\rule{thinmathspace}{0ex}}\Vert f{\Vert}_{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\rho (z,w{)}^{2}}\end{array}$$(15)

and

$$\begin{array}{}{\displaystyle |f(z)-f(w)-[{f}^{\prime}(z)(1-|z{|}^{2})+{f}^{\prime}(w)(1-|w{|}^{2})]\frac{\psi (z,w)}{2}|\le c\phantom{\rule{thinmathspace}{0ex}}\Vert f{\Vert}_{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\rho (z,w{)}^{3}.}\end{array}$$(16)

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

$$\begin{array}{}{\displaystyle |{w}_{n+1}^{\prime}|(1-|{z}_{n+1}{|}^{2})\le |{f}^{\prime}({z}_{n+1})(1-|{z}_{n+1}{|}^{2})-{f}^{\prime}({z}_{n})(1-|{z}_{n}{|}^{2})|+|{f}^{\prime}({z}_{n})|(1-|{z}_{n}{|}^{2})|1-{a}_{n}^{\prime}|.}\end{array}$$

If on the right of this last inequality, we put

$$\begin{array}{}{\displaystyle |{f}^{\prime}({z}_{n+1})(1-|{z}_{n+1}{|}^{2})+{f}^{\prime}({z}_{n})(1-|{z}_{n}{|}^{2})|+|{f}^{\prime}({z}_{n})|(1-|{z}_{n}{|}^{2})|1+{a}_{n}^{\prime}|,}\end{array}$$

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*

$$\begin{array}{}{\displaystyle |{B}_{m+1}({z}_{m+1})|\ge c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Gamma}({a}_{m})\phantom{\rule{1em}{0ex}}\mathrm{\forall}m\in \mathbb{N},}\end{array}$$

*and condition* (D) *if*

$$\begin{array}{}{\displaystyle |{B}_{m+1}({z}_{m+1})|\ge c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Pi}({a}_{m})\phantom{\rule{1em}{0ex}}\mathrm{\forall}m\in \mathbb{N}.}\end{array}$$

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

$$\begin{array}{}{\displaystyle |{B}_{m+1}({z}_{m+1}){|}^{2}\ge c\phantom{\rule{thinmathspace}{0ex}}\mathit{\Lambda}({a}_{m}^{\prime})\phantom{\rule{1em}{0ex}}\mathrm{\forall}m\in \mathbb{N}.}\end{array}$$

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.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.