1 Background and the main results
1.1 Cyclic singular inner functions
Let X be a topological space consisting of functions which are analytic in the unit disk $\mathbb {D} = \{ z \in \mathbb {C} : z < 1\}$ and which satisfy some customary desirable properties, such as that the evaluation $f \mapsto f(\lambda )$ is a continuous functional on X for each $\lambda \in \mathbb {D}$ and that the function $z \mapsto zf(z)$ is contained in the space X whenever $f \in X$ . A function $g \in X$ is said to be cyclic if there exists a sequence of analytic polynomials $\{p_n\}_n$ for which the polynomial multiples $\{gp_n\}_n$ converge to the constant function $1$ in the topology of the space.
The wellknown Hardy classes $H^p$ are among the very few examples of analytic function spaces in which the cyclicity phenomenon is completely understood. The cyclic functions g are of the form
where $d\textit {m}$ is the (normalized) Lebesgue measure of the unit circle $\mathbb {T} = \{ z\in \mathbb {C} : z = 1\}$ . Functions as in (1.1) are called outer functions. The inner functions are of the form
where $\nu $ is a positive finite singular Borel measure on $\mathbb {T}$ and $\{\alpha _n\}_n$ is a Blaschke sequence. It is clear that if the Blaschke product B on the left is nontrivial, then $\theta $ vanishes at points in $\mathbb {D}$ and therefore cannot be cyclic in any reasonable space of analytic functions X. The right factor $S_\nu $ is a singular inner function, and it is well known that if a function $g \in H^p$ has a singular inner function as a factor, then g is not cyclic in $H^p$ . As a consequence, if $\{p_n\}_n$ is a sequence of polynomials for which we have
then necessarily the Hardy class norms of the sequence must explode
for finite $p \geq 1$ , or in case $p = \infty $ ,
When other norms are considered, cyclic singular inner functions might exist, and here the Bergman spaces $L^p_a(\mathbb {D})$ provide a famous set of examples. The Bergman norms are of the form
where $dA$ is the normalized area measure of $\mathbb {D}$ . After a sequence of partial results by multiple authors, Korenblum in [Reference Korenblum12] and Roberts in [Reference Roberts15] independently characterized the cyclic singular inner functions in the Bergman spaces in terms of the vanishing on certain subsets of $\mathbb {T}$ of the corresponding singular measure $\nu $ appearing in (1.2). A construction of a singular inner function which is cyclic in the classical Bloch space appears in [Reference Anderson, Fernandez and Shields3].
Recently, Ransford in [Reference Ransford14] noted that singular inner functions exist which decay arbitrarily slowly near the boundary of the disk. As we shall see below, this fact has as a direct consequence the existence of an abundance of spaces of analytic function which admit cyclic singular inner functions. Here is the precise statement of the main result of [Reference Ransford14].
Theorem 1.1 Let $w:[0,1) \to (0,1)$ be any function satisfying $\lim _{r \to 1^} w(r) = 0$ . Then there exists a singular inner function $\theta $ for which we have
It has been remarked to the present author that, in fact, this theorem appears already in the literature. For instance, Shapiro similarly mentions in [Reference Shapiro17] that a singular inner function always satisfies an estimate of the form
where C is some positive constant, and $\omega = \omega _\nu $ is the modulus of continuity of the measure $\nu $ :
The supremum above is taken over arcs I of the circle $\mathbb {T}$ which are of length h. In [Reference Shapiro18], Shapiro proves that a singular measure $\nu $ exists with a modulus of continuity $\omega _\nu $ for which $\omega _\nu (h)/h$ grows to infinity arbitrarily slowly as h decreases to zero, hence proving Theorem 1.1 as a consequence of the estimate (1.4). In fact, such singular measures have been known to exist at least since the work of Hartman and Kershner in [Reference Hartman and Kershner10]. The proof of Ransford in [Reference Ransford14] also involves establishing the existence of such a measure.
The following result is the abovementioned consequence of Theorem 1.1 on existence of cyclic singular inner function. The result is surely well known, and has an elementary proof which we include for convenience.
Corollary 1.2 Let $w:[0,1) \to (0,1)$ be any decreasing function satisfying $\lim _{t \to 1^} w(t) = 0$ . There exist a singular inner function $\theta = S_\nu $ and a sequence of analytic polynomials $\{p_n\}_n$ such that:

(1) $\lim _{n \to \infty } \theta (z) p_n(z) = 1, \quad z \in \mathbb {D},$

(2) $\sup _{z \in \mathbb {D}} \theta (z) p_n(z) w(z) \leq 2$ .
Proof Apply Theorem 1.1 to the function w to produce a singular inner function $\theta $ satisfying (1.3). For integers $n \geq 2$ , we set $r_n := 1 1/n$ and $Q_n(z) := 1/\theta (r_n z)$ . Then $Q_n$ is holomorphic in a neighborhood of the closed disk $\overline {\mathbb {D}}$ , and because we are assuming that w is decreasing, we have the estimate
We can approximate $Q_n$ by an analytic polynomial $p_n$ so that
Then
It is clear from the construction that $\theta (z)p_n(z) \to 1$ as $n \to \infty $ , for any $z \in \mathbb {D}$ .
Corollary 1.2 says that there exist cyclic singular inner functions in essentially any space of analytic functions defined in terms of a growth condition, or in any space in which such a growth space is continuously embedded.
The purpose of this note is to apply Theorem 1.1, or more precisely its simple consequence stated in Corollary 1.2, to the questions of existence of functions with certain smoothness properties in model spaces $K_\theta $ . We will establish sharpness of certain existing approximation results in these spaces. Moreover, we take the opportunity to discuss similar questions in the broader class of de Branges–Rovnyak spaces $\mathcal {H}(b)$ . Our results are proved by rather wellknown methods, but their statements seem to be missing in the existing literature, and we wish to fill in this gap.
In the proofs of the main results, which will be stated shortly, we will concern ourselves with the following weak type of cyclicity of singular inner functions. Let Y be some linear space of analytic functions which is contained in $H^1$ . We want to investigate if there exist a singular inner function $\theta $ and a sequence of polynomials $\{p_n\}_n$ such that
holds for all $f \in Y$ . The above situation means that the sequence $\{\theta p_n\}_n$ converges to the constant $1$ , weakly over the space Y. Now, clearly, if Y is too large of a space (say, $Y = H^2$ ), then (1.6) can never hold for all $f \in Y$ . However, if Y is sufficiently small, then the situation in (1.6) might occur. For instance, in the extreme case, when Y is a set of analytic polynomials, then any singular inner function $\theta $ and any sequence of polynomials $\{p_n\}_n$ which satisfies $\lim _{n \to \infty } p_n(z) = 1/\theta (z)$ for $z \in \mathbb {D}$ is sufficient to make (1.6) hold. Philosophically speaking, it is the uniform smoothness of the functions in the class Y that allows the existence of singular inner functions $\theta $ for which the above situation occurs. Under insignificant assumptions on Y, a straightforward argument shows that if (1.6) occurs, then the intersection between Y and $K_\theta $ is trivial, whereas Corollary 1.2 provides us with a huge class of spaces Y for which (1.6) can be achieved.
1.2 Main results
Recall that the space $K_\theta $ is constructed from an inner function $\theta $ by taking the orthogonal complement of the subspace
in the Hardy space $H^2$ :
For background on the spaces $K_\theta $ , one can consult the books [Reference Cima, Matheson and Ross5, Reference Garcia, Mashreghi and Ross9]. In our first result, we will show that the famous approximation theorem of Aleksandrov from [Reference Aleksandrov1] on density in $K_\theta $ of functions which extend continuously to the boundary is in fact essentially sharp, as it cannot be extended to any class of functions satisfying an estimate on their modulus of continuity. By a modulus of continuity $\omega $ , we mean here a function $\omega : [0,\infty ) \to [0, \infty )$ which is continuous, increasing, satisfies $\omega (0) = 0$ , and for which $\omega (t)/t$ is a decreasing function with
For such a function $\omega $ , we define $\Lambda ^\omega _a$ to be the space of functions f which are analytic in $\mathbb {D}$ , extend continuously to $\overline {\mathbb {D}}$ , and satisfy
Then $\Lambda ^\omega _a$ is the space of analytic functions on $\mathbb {D}$ which have a modulus of continuity dominated by $\omega $ . We make $\Lambda ^\omega _a$ into a normed space by introducing the quantity
By a theorem of Tamrazov from [Reference Tamrazov19], we could have replaced the supremum over $\overline {\mathbb {D}}$ by a supremum over $\mathbb {T}$ , and obtain the same space of functions (we remark that a nice proof of this result is contained in [Reference Bouya4, Appendix A]). The following is an optimality statement regarding Aleksandrov’s density theorem.
Theorem 1.3 Let $\omega $ be a modulus of continuity. There exists a singular inner function $\theta $ such that
This statement will be proved in Section 3. In fact, we will see that Theorem 1.3 is a consequence of a variant, and in some directions a strengthening, of a theorem of Dyakonov and Khavinson from [Reference Dyakonov and Khavinson6]. For a sequence of positive numbers $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ , we define the class
The next theorem, proved in Section 2, reads as follows.
Theorem 1.4 Let $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ be any increasing sequence of positive numbers with $\lim _{n \to \infty } \lambda _n = \infty $ . Then there exists a singular inner function $\theta $ such that
The result can be compared to the mentioned result of Dyakonov and Khavinson in [Reference Dyakonov and Khavinson6], from which the above result can be deduced in the special case $\boldsymbol {\lambda } = \{ (k+1)^{\alpha }\}_{k=0}^\infty $ with any $\alpha> 0$ .
The theory of de Branges–Rovnyak spaces $\mathcal {H}(b)$ is a wellknown generalization of the theory of model spaces $K_\theta $ . The symbol of the space b is now any analytic selfmap of the unit disk, and we have $\mathcal {H}(b) = K_b$ whenever b is inner. For background on $\mathcal {H}(b)$ spaces, one can consult [Reference Sarason16] or [Reference Fricain and Mashreghi7, Reference Fricain and Mashreghi8]. A consequence of the author’s work in collaboration with Aleman in [Reference Aleman and Malman2] is that the abovementioned density theorem of Aleksandrov generalizes to the broader class of $\mathcal {H}(b)$ spaces: any such space admits a dense subset of functions which extend continuously to the boundary. Since Theorem 1.3 proves optimality of Aleksandrov’s theorem for inner functions $\theta $ , one could ask if at least for outer symbols b any improvement of the density result in $\mathcal {H}(b)$ from [Reference Aleman and Malman2] can be obtained. In Section 4, we remark that this is not the case, and the result in [Reference Aleman and Malman2] is also essentially optimal, even for outer symbols b.
Theorem 1.5 Let $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ be any increasing sequence of positive numbers with $\lim _{n \to \infty } \lambda _n = \infty $ . There exists an outer function $b:\mathbb {D} \to \mathbb {D}$ such that
Theorem 1.6 Let $\omega $ be a modulus of continuity. There exists an outer function $b:\mathbb {D} \to \mathbb {D}$ such that
We will show that the above results are essentially equivalent to a theorem of Khrushchev from [Reference Khrushchev11].
In Section 5, we list a few questions we have not found an answer for, and some ideas for further research.
2 Proof of Theorem 1.4
In the proof of the theorem, we will need to use the following crude construction of an integrable weight with large moments.
Lemma 2.1 Let $\{\lambda _n\}_{n=0}^\infty $ be a decreasing sequence of positive numbers with $\lim _{n \to \infty } \lambda _n = 0$ . There exists a nonnegative function $\Lambda \in L^1([0,1])$ which satisfies
Proof Recall that the sequence $(11/n)^n = \exp (n\log (11/n))$ is decreasing and satisfies
It follows that
for some constant $\alpha> 0$ which is independent of n. For $n \geq 1$ , we define the intervals $I_n = \big ( 1  1/n, 1  1/(n+1) \big )$ . Our function $\Lambda $ will be chosen to be of the form
where $1_{I_n}$ is the indicator function of the interval $I_n$ and the $c_n$ are positive constants to be chosen shortly. Note that
We choose
This choice of coefficients $c_n$ makes $\Lambda $ integrable over $[0,1]$ :
In the last step, we used the assumption that the sequence $\{\lambda _n\}_n$ converges to zero. Moreover, by (2.1) and the choice of $c_n$ , we can estimate
The proof is complete.
The significance of the above lemma is the estimate
for some numerical constant $c> 0$ and any function f which is holomorphic in a neighborhood of the closed disk $\overline {\mathbb {D}}$ . The estimate can be verified by direct computation of the integral on the righthand side, using polar coordinates.
We will also use the following wellknown construction.
Lemma 2.2 For any function $g \in L^1([0,1])$ , there exists a positive and increasing function $w: [0,1) \to \mathbb {R}$ which satisfies
and
Proof The integrability condition on g implies that
Thus, there exists a sequence of intervals $\{I_n\}_{n=1}^\infty $ which have $1$ as the right endpoint and length shrinking to zero, which satisfy $I_{n+1} \subset I_n$ for all $n \geq 1$ , and
If we set
where $1_{I_n \setminus I_{n+1}}$ is the indicator function of the set difference $I_n \setminus I_{n+1}$ , then w is increasing, satisfies $\lim _{t \to 1^} w(t) = \infty $ , and
for all $n \geq 1$ . Consequently,
Proof of Theorem 1.4
Let $\Lambda $ be the function in Lemma 2.1 which corresponds to the sequence $\{1/\lambda _n\}_{n=0}^\infty $ . That is, $\Lambda $ satisfies
and $\Lambda \in L^1[0,1]$ . Now, let w be a positive decreasing function which satisfies $w(x) < 1/2$ , $\lim _{x \to 1^} w(x) = 0$ and
Existence of such a function follows readily from Lemma 2.2. Apply Corollary 1.2 to w and obtain a corresponding inner function $\theta $ and a sequence of polynomials $\{p_n\}_n$ for which the conclusions (i) and (ii) of Corollary 1.2 hold. We will show that for this $\theta $ , we have $K_\theta \cap H^2_{\boldsymbol {\lambda }} = \{0\}.$
Indeed, assume that $f \in K_\theta \cap H^2_{\boldsymbol {\lambda }} = \{0\},$ but that in fact f is nonzero. Since both $K_\theta $ and $H^2_{\boldsymbol {\lambda }}$ are invariant for the backward shift operator, we may without loss of generality assume that $f(0) \neq 0$ . Fix an integer n, and let
Let $\{f_k\}_k, \{g_k\}_k$ be the sequences of Taylor coefficients of f and g, respectively. Since $f \in K_\theta $ , we have
Using inequality (2.2) on the term on the righthand side in the last expression (with $\lambda _n$ replaced by $1/\lambda _n$ ), we obtain
By assertion in part (ii) of Corollary 1.2, the function $g(z)^2\Lambda (z)$ is dominated pointwise in $\mathbb {D}$ by the integrable function
independently of which polynomial $p_n$ is used to defined g in (2.3). However, if we let $n \to \infty $ in (2.3), then $g(z)^2\Lambda (z) \to 0$ , and so we infer from the computation above and the dominated convergence theorem that $f(0) = 0$ , which is a contradiction. The conclusion is that $K_\theta \cap H^2_{\boldsymbol {\lambda }} = \{ 0\}$ , and the proof of the theorem is complete.
3 Proof of Theorem 1.3
Theorem 1.3 will follow immediately from Theorem 1.4 together with the following embedding result for the spaces $\Lambda ^\omega _a$ .
Lemma 3.1 Let $\omega $ be a modulus of continuity. There exists an increasing sequence of positive numbers $\boldsymbol {\alpha } = \{ \alpha _n\}_{n=0}^\infty $ satisfying $\lim _{n \to \infty } \alpha _n = \infty $ such that for any $f \in \Lambda ^\omega _a$ we have the estimate
where $C> 0$ is a numerical constant and $\{f_n\}_n$ is the sequence of Taylor coefficients of f.
Proof For each $r \in (0,1)$ , we have the estimate
Since $\lim _{t \to 0} \omega (t) = 0$ , for each positive integer N, there exists a number $r_N \in (0,1)$ such that $\omega (1r_N) \leq \frac {1}{2^N}$ . Since $\lim _{n \to \infty } r_N^{2n} = 0$ , there exists an integer $K(N)$ such that $r_N^{2n} < 1/2$ for $n \geq K(N)$ . Then
Consequently,
We can clearly choose the sequence of integers $K(N)$ to be increasing with N. If we define the sequence $\boldsymbol {\alpha }$ by the equation $\alpha _n = 1$ for $n < K(1)$ , and $\alpha _n = 2^N$ for $K(N) \leq n < K(N+1)$ , then (3.1) follows readily from (3.3) by summing over all $N \geq 1$ .
Proof of Theorem 1.3
Lemma 3.1 implies that $\Lambda ^\omega _a$ is contained in some space of the form $H^2_{\boldsymbol {\alpha }}$ as defined in (1.8). If $\theta $ is a singular inner function given by Theorem 1.4 such that $H^2_{\boldsymbol {\alpha }} \cap K_\theta = \{0\}$ , then obviously we also have that $\Lambda ^\omega _a \cap K_\theta = \{0\}$ , and so the claim follows.
4 Proofs of Theorems 1.5 and 1.6
Here, we prove the optimality of the continuous approximation theorem for the larger class of $\mathcal {H}(b)$ spaces. As remarked in the introduction, this is essentially equivalent to a theorem of Khrushchev from [Reference Khrushchev11].
Proof of Theorem 1.5
By a result of Khrushchev noted in [Reference Khrushchev11, Theorem 2.4], there exists a closed subset E of the circle $\mathbb {T}$ with the property that for no nonzero integrable function h supported on E is the Cauchy integral
a member of the space $H^2_{\boldsymbol {\lambda }}$ . It suffices thus to construct an $\mathcal {H}(b)$ space for which every function can be expressed as such a Cauchy integral. The simplest choice for the space symbol b is the outer function with modulus $1$ on $\mathbb {T} \setminus E$ and $1/2$ on E. Then b is invertible in the algebra $H^\infty $ , and a consequence of the general theory (see [Reference Fricain and Mashreghi8, Theorems 20.1 and 28.1]) is that every function in the space $\mathcal {H}(b)$ is a Cauchy integral of a function h which is squareintegrable on $\mathbb {T}$ and supported only on E. Thus, $H^2_{\boldsymbol {\lambda }} \cap \mathcal {H}(b) = \{ 0 \}$ , by Khrushchev’s theorem.
Finally, Theorem 1.6 follows from Theorem 1.5 in the same way as Theorem 1.3 follows from Theorem 1.4.
5 Some ending questions and remarks
Since Theorem 1.1 seems to be such a powerful tool in establishing results of the kind mentioned here, we are wondering whether it can be further applied. In particular, the following questions come to mind.

(1) Are our methods strong enough to prove that there exist model spaces $K_\theta $ which admit no nonzero functions in the Wiener algebra of absolutely convergent Fourier series? The result is known, and has been noted in [Reference Limani and Malman13]. However, it was proved as a consequence of a complicated construction of a cyclic singular inner function in the Bloch space. Is it so that the construction in Corollary 1.2 is sufficient to prove the nondensity result for the Wiener algebra in the fashion presented here?

(2) For $p> 2$ , the Banach spaces $\ell ^p_a$ consisting of functions $f\in Hol(\mathbb {D})$ with Taylor series $\{f_n\}_{n=0}^\infty $ satisfying
$$ \begin{align*}\f\^p_{\ell^p_a} := \sum_{n=0}^\infty f_n^p < \infty\end{align*} $$are of course larger than the space $H^2 = \ell ^2_a$ . Do there exist cyclic singular inner functions in these spaces?
Acknowledgment
The author would like to thank Christopher Felder for his reading of the manuscript and for his helpful suggestions for improvements. He would also like to thank Adem Limani for very useful discussions and important comments.