TOPOLOGIES ON THE SMIRNOV CLASS

John E. McCarthy

Abstract. The Smirnov Class N^{+} of analytic functions on the disk has traditionally been studied with the complete metric topology given by

ρ(f, g) = sup_{0<r<1} \frac{1}{2π} ∫_{𝕋} log(1 + |f(re^{iθ}) − g(re^{iθ})|)dθ.

However, the space can be represented as the union of certain weighted Hardy spaces H^{2}(w) (the closure of the polynomials in L^{2}(wdθ)), and hence given the inductive limit topology H. We give a new proof of Yanagihara’s characterisation of the dual of (N^{+}, ρ). We prove that (N^{+}, H) is an (incomplete) metrizable topological algebra in which Fourier series converge. We then study the individual spaces H^{2}(w), prove asymptotic versions of Szegö’s theorem and the Helson-Szegö theorem on these spaces, and characterise the universal multipliers of their duals.

0. Introduction

The Smirnov class N^{+} consists of those functions f analytic on the unit disk 𝔻 for

which

lim_{r→1} \frac{1}{2π} ∫_{𝕋} log(1 + |f(re^{iθ})|)dθ = \frac{1}{2π} ∫_{𝕋} log(1 + |f(e^{iθ})|)dθ < ∞.

(The boundedness of the integrals on the left implies that f is in the Nevanlinna class, and

so has non-tangential limits almost everywhere on the circle. It is this boundary function

that we mean in the second integral.) We shall define ∥f∥ by

∥f∥ = sup_{0<r<1} \frac{1}{2π} ∫_{𝕋} log(1 + |f(re^{iθ})|)dθ.

This is not a norm, but it does define a metric (ρ(f, g) := ∥f − g∥), that makes N^{+} into

a topological vector space. For basic facts about N^{+}, see e.g [4], [7] or [18].

In [22], N. Yanagihara found asymptotic bounds on the growth of the Taylor coefficients

of a function in N^{+}, and in [23], he used these to characterise the multipliers of N^{+} into

H^{∞}, and hence to find the dual of (N^{+}, ρ). The first two results have recently been

Typeset by AMS-TEX 2 JOHN E. MCCARTHY

generalised by M. Nawrocki [16] to Banach space valued functions. We give simple proofs of Yanagihara’s theorems in Section 1.

The Nevanlinna brothers proved that a function is in N^{+} if and only if it can be expressed as the ratio g/h, where g and h are in H^{2}, the classical Hardy space, and h is outer ([17], or see [4]). But

f = \frac{g}{h} ⟺ hf ∈ H^{2}
⟺ inf{∫_{𝕋} |hf − p|^{2}dθ : p a polynomial} = 0
⟺ inf{∫_{𝕋} |hf − hp|^{2}dθ : p a polynomial} = 0

(since h is outer, so {hp} is dense in H^{2})

⟺ inf{∫_{𝕋} |f − p|^{2}|h|^{2}dθ : p a polynomial} = 0
⟺ f ∈ H^{2}(|h|^{2}),

where H^{2}(|h|^{2}) denotes the closure of the polynomials in L^{2}(|h|^{2}dθ). Thus

N^{+} = ⋃_{h outer} H^{2}(|h|^{2}).

There are two topologies associated with this representation of N^{+}: the usual locally convex inductive limit topology, which we shall call the Helson topology and denote H, in which a neighbourhood base for 0 is given by those balanced convex sets whose intersection with each H^{2}(|h|^{2}) is a neighbourhood of zero in H^{2}(|h|^{2}); and a not locally convex topology, which we shall denote I, in which a neighbourhood base for zero is given by all sets whose intersection with each H^{2}(|h|^{2}) is a neighbourhood of zero. It is not immediately obvious that I is strictly stronger than H; but in [15], it was proved that I coincides with the metric topology ρ, which is not locally convex.

This sheds a lot of light on H; for a start, as H is the finest locally convex topology coarser than I, (N^{+}, H) and (N^{+}, I) must have the same duals. It is a theorem of Helson that the dual of (N^{+}, H) can be identified with those functions that are in the range

TOPOLOGIES ON THE SMIRNOV CLASS 3

of every Toeplitz operator with co-analytic symbol [8]; so, combined with Yanagihara’s characterisation of (N^{+}, ρ)^{∗} (Theorem 1.7 below), one gets that the common range of these Toeplitz operators are those functions whose n^{th} Fourier coefficients are O(e^{−c\sqrt{n}}), for some positive c; see [15].

As H is Hausdorff, and I is metrizable, H too must be metrizable; in fact the metric can be given explicitly, as Yanagihara did in [24]:

d(f(z) ∼ ∑_{n=0}^{∞} a_{n}z^{n}, g(z) ∼ ∑_{n=0}^{∞} b_{n}z^{n}) = ∑_{k=1}^{∞} 2^{−k} \frac{∑_{n=0}^{∞} |a_{n} − b_{n}|e^{−\sqrt{n/k}}}{1 + ∑_{n=0}^{∞} |a_{n} − b_{n}|e^{−\sqrt{n/k}}} .  (0.1)

We discuss this in Section 2. Multiplication with respect to this metric is continuous [25], so it also follows that (N^{+}, H) is a topological algebra. In [9] Helson showed that multiplication is separately continuous, directly from the definition of H, but proving joint continuity seems to require passing to the more tractable metric topology d in (0.1).

In Section 2 we also show that Fourier series converge in H (strictly speaking, we should talk about power series, because the functions are not necessarily summable on the unit circle; but as the series for an H^{2} function is always called a Fourier series, we shall use this terminology). Observe that for any weight which is not a Helson-Szegö weight, for example |e^{iθ} − 1|^{2}dθ, Fourier series do not always converge in H^{2}(w) [10]. Indeed the Fourier series of 1/(1 − z) does not converge in any H^{2}(w), nor in (N^{+}, I). That the imposition of local convexity on the inductive limit makes the divergences disappear seems to reflect, in some vague sense, the symmetry of divergent Fourier series.

In Section 3 we consider the individual spaces H^{2}(w), the closure of the (analytic) polynomials in L^{2}(wdθ), where w is a positive integrable function. Letting W be the set of weights w for which ∫_{𝕋} log(w(e^{iθ}))dθ > −∞, it is a famous result of Szegö that evaluating a polynomial at zero extends to be a continuous linear functional on H^{2}(w) if and only if w is in W. If this holds, it is easy to show that evaluating the n^{th} Taylor coefficient at zero is also a continuous functional, and that the norms of these functionals grow more slowly than e^{cn}, for any positive c. We prove that the bound can actually be replaced by e^{c\sqrt{n}}, and that this cannot be improved for general w (Theorem 3.1). We also prove an asymptotic Helson-Szegö theorem, on the angle between the subspace (in

4 JOHN E. MCCARTHY

L^{2}(wdθ)) of analytic polynomials of degree less than n, and the space of all co-analytic polynomials with no constant term - namely that the sine of this angle must decrease more slowly than e^{−c\sqrt{n}} for every positive c.

The principal result of Section 3 is the proof that the set of universal multipliers of the duals of all the H^{2}(w)’s, as w ranges over W, is again the set of holomorphic functions whose Fourier coefficients decay like e^{−c\sqrt{n}}. Let us briefly explain its significance.

Let b be a function in the unit ball of H^{∞}, the space of bounded analytic functions in the ball. The de Branges space H(b) is defined to be the range of the operator (1−T_{b}Tb̄)^{1/2}. These spaces, originally introduced in [1], have been the object of much study recently. One question of particular interest is to determine, for fixed b, what functions multiply H(b) into itself - see e.g. [14], [20]. The question splits into two cases, depending on whether b is an extreme point of the unit ball of H^{∞} (and no non-constant polynomial is a multiplier), or it isn’t (and every polynomial is a multiplier) [20]. In [2], it is shown that a function m is a multiplier of every H(b), for b not an extreme point of the unit ball of H^{∞}, if and only if m is a multiplier of H^{2}(w)^{∗} for every w in W; and hence Theorem 3.2 shows that these hyper-smooth functions surface again in a different context.

We would like to thank Mark Davis and Henry Helson for their valuable suggestions.

1. The dual of (N^{+}, ρ)

We shall need three lemmata, the proofs of which are elementary, and can be found in [18, II.3.1 and II.11.2].

Lemma 1.1. If f is in N^{+} and z is in 𝔻, then

|f(z)| ≤ e^{\frac{2∥f∥}{1−|z|}} − 1.

Lemma 1.2. Let f be in N^{+}. Then its Taylor coefficients satisfy

|f̂(n)| ≤ e^{\sqrt{8n∥f∥}(1+ϵ_{n})},

where ϵ_{n} is o(1), depends only on ∥f∥, and decreases as ∥f∥ decreases.

TOPOLOGIES ON THE SMIRNOV CLASS 5

Lemma 1.3. Let c be positive, and define f_{c}(z) = exp(\frac{c}{8} \frac{1+z}{1−z}) − 1. Then f̂_{c}(n) = e^{c\sqrt{n}(1+o(1))}.

We first prove Yanagihara’s theorem on the growth of Taylor coefficients [22].

Theorem 1.4. Let f be in N^{+}, and let c > 0. Then f̂(n) = O(e^{c\sqrt{n}}).

Proof. By the monotone convergence theorem, lim_{t→0} ∥tf∥ = 0. So choose t so that \sqrt{8∥tf∥} ≤ \frac{c}{2}, and apply Lemma 1.2. □

Before giving our proof of Yanagihara’s theorem on the dual of (N^{+}, ρ), we first need one more lemma, about the size of the functions f_{c} from Lemma 1.3.

Lemma 1.5. lim_{c↓0} ∥f_{c}∥ = 0.

Proof. Let ϵ > 0, and let δ > 0 be such that if |w| < δ, then |e^{w} − 1| < ϵ. Let B_{c} = {z ∈ 𝔻 : |_{\overline{↚}} \frac{⊮+𝟋}{⊮−𝟋}| ≥ δ\}.

For any 0 < r < 1, the integral

\frac{1}{2π} ∫_{𝕋} log(1 + |f_{c}(re^{iθ})|)dθ = \frac{1}{2π} ∫_{𝕋} log(1 + |exp\frac{c}{8} \frac{1 + re^{iθ}}{1 − re^{iθ}} − 1|)dθ  (1.6)

decomposes into the integral over the arc where re^{iθ} is not in B_{c}, which can contribute at most log(1 + ϵ), and the complementary arc, I(c, r).

But the integrand is always less than or equal to 2+ℜ\frac{c}{8} \frac{1+re^{iθ}}{1−re^{iθ}}, so the contribution from I(c, r) is at most \frac{1}{π} |I(c, r)| + \frac{c}{8}. As c decreases to zero, |I(c, r)| gets small uniformly in r, so c can be chosen to make all the integrals in (1.6), for all 0 < r < 1, arbitrarily small. □

To characterise the dual of (N^{+}, ρ), notice that, because f(rz) tends to f(z) as r increases to 1, the polynomials are dense in N^{+}, and so any continuous linear functional is uniquely defined by knowing its action on the monomials z^{n}. We shall let A denote the disk algebra, those functions analytic on 𝔻 and continuous on the closure.

Theorem 1.7. Let Γ be a linear functional, defined on polynomials by Γ(∑_{n} a_{n}z^{n}) = ∑_{n} a_{n}γ_{n}. Then Γ extends to be continuous on (N^{+}, ρ) if and only if, for some c > 0, γ_{n} = O(e^{−c\sqrt{n}}).

6 JOHN E. MCCARTHY

Proof. (Sufficiency) By Theorem 1.4, Γ(f(z) ∼ ∑_{n=0}^{∞} a_{n}z^{n}) := ∑_{n=0}^{∞} a_{n}γ_{n} defines a linear functional on all of N^{+}. Moreover it is continuous, because if f_{k} tends to zero, and ϵ > 0 is given, choose N large enough that ∑_{n=N}^{∞} e^{−c\sqrt{n}/2} < \frac{ϵ′}{2} , where ϵ′ = ϵ/ sup_{n}\{γ_{n}e^{c\sqrt{n}}\}; and choose K large enough that, for k ≥ K, if n ≥ N then |f̂_{k}(n)| ≤ e^{c\sqrt{n}/2} (can do this by Lemma 1.2), and if n < N then |f̂_{k}(n)| ≤ \frac{ϵ′}{2N} (can do this by Lemma 1.1). Then |Γ(f_{k})| ≤ ϵ, for k ≥ K.

(Necessity) Define a map T from polynomials into A by T (∑_{n=0}^{∞} a_{n}z^{n}) = ∑_{n=0}^{∞} a_{n}γ_{n}z^{n}. The norm of T (p(z)) in the disk algebra is

∥T (p(z))∥_{A} = sup_{|λ|=1} |Γ(p(λz))|.

As ∥.∥ is rotation invariant, and Γ is continuous, T is a continuous map from N^{+} into A; in particular, there is some d > 0 such that the ball of radius d in N^{+} gets mapped into the ball of radius 1 in A. Therefore we can conclude that if f is in N^{+}, and ∥f∥ < d, then |f̂(n)γ_{n}| ≤ 1.

Now, consider the functions f_{c} of Lemma 1.3. These functions are not in N^{+}, but f_{c}(rz) is in N^{+} for all 0 ≤ r < 1, and, by Lemma 1.5, c can be chosen so that ∥f_{c}(rz)∥ < d for all r in this range. Letting r = 2^{−1/n}, we get

|γ_{n}| ≤ \frac{1}{r^{n}} e^{−c\sqrt{n}(1+o(1))} = O(e^{−c\sqrt{n}}),

as required. □

It is easy to show that (N^{+}, ρ) is actually complete (see e.g. [6, p.123] or [23]). This makes the proof of sufficiency above even easier - just apply the uniform boundedness principle [3, p.52] to the partial sums. The uniform boundedness principle also proves that the bound in Theorem 4 cannot be replaced by O(e^{c_{n}\sqrt{n}}) for any sequence c_{n} decreasing to zero, for otherwise Γ(z^{n}) = \frac{1}{(n+1)^{2}} e^{−c_{n}\sqrt{n}} would extend to a continuous linear functional that doesn’t satisfy Theorem 5. Finally, the same ideas yield that if a sequence {γ_{n}}_{n=0}^{∞} multiplies N^{+} into H^{p} for some p > 0 (i.e. ∑_{n=0}^{∞} a_{n}z^{n} in N^{+} implies ∑_{n=0}^{∞} a_{n}γ_{n}z^{n} is in H^{p}), then, because Taylor coefficients of functions in H^{p} can grow only polynomially fast [4, p.98], \{γ_{n}\}_{n=0}^{∞} actually multiplies N^{+} into A^{∞}, and must decay like e^{−c\sqrt{n}}.

TOPOLOGIES ON THE SMIRNOV CLASS 7

2. Convergence of Fourier series in H

In [24] Yanagihara studied the space F^{+}, which consists of those functions f(z) ∼ ∑_{k=0}^{∞} a_{k}z^{k} for which a_{k} = O(exp(o\sqrt{k})), topologised by the family of norms

∥f∥_{c} = ∑_{k=0}^{∞} |a_{k}|e^{−c\sqrt{k}},

where c is a positive constant. This can be made into a metric topology by putting

d(f, g) = ∑_{n=1}^{∞} 2^{−n} \frac{∥f − g∥_{1/\sqrt{n}}}{1 + ∥f − g∥_{1/\sqrt{n}}} .

With respect to this topology F^{+} is a complete, locally convex metric space, i.e. a Fréchet space. He showed that F^{+} is actually the containing Fréchet space of (N^{+}, ρ), i.e. (N^{+}, ρ) embeds densely into F^{+}, under the natural inclusion, and (F^{+}, d)^{∗} = (N^{+}, ρ)^{∗}.

As d is weaker than ρ, (N^{+}, d)^{∗} = (N^{+}, ρ)^{∗}. So H and d are both locally convex topologies on N^{+} that give rise to the same dual. Now, on any space there is a strongest locally convex topology with a given dual, called the Mackey topology for that dual pair. Certain conditions on a topology force it to coincide with the Mackey topology; in particular, being barrelled, as H is, or being metrizable, as d is, both make the topologies coincide with the Mackey topology [21, 10-1.9],so H and d must be the same. In particular, the Helson topology must be metrizable, and so have a countable neighborhood base, and d must be barrelled.

(Recall that a locally convex topological vector space is called barrelled if every closed, absolutely convex, absorbing set is a neighborhood of zero. Every Banach space is barrelled, and the inductive limit of barrelled spaces is barrelled, so (N^{+}, H) is barrelled. The importance of being barrelled is that every pointwise bounded family of continuous linear functionals is equicontinuous. See e.g [21, 9-3.4] or [12, p. 141].)

We shall say that a topology on N^{+} is Szegö if the functionals g ↦ ĝ(k) are continuous (equivalently, the forward and backward shifts are continuous). All natural topologies on N^{+} are Szegö.

8 JOHN E. MCCARTHY

Theorem 2.1. Let τ be a barrelled Szegö topology on N^{+}. Then the following are equivalent:

(i) τ = H (ii) τ = d (iii) Fourier series converge in (N^{+}, τ)

Proof. (ii) ⟺ (i) follows from the above remarks.

(i) ⟹ (iii): Let g ∼ ∑_{k=0}^{∞} a_{k}z^{k} be in N^{+}. Set

R_{N} = N ∑_{k=N+1}^{∞} a_{k}z^{k}.

Let Φ be in (N^{+}, H)^{∗}. By [15], Φ(z^{k}) = O(e^{−c\sqrt{k}}), for some c > 0. Therefore by Theorem 1.4 (applied to c/2) the set

{Φ(R_{N})}_{N=0}^{∞}

is bounded, so the set {R_{N}} is weakly bounded. Because (N^{+}, H) is locally convex, the set \{R_{N}\} must therefore be H-bounded [19,p. 68]. So \frac{1}{N} R_{N} must tend to zero (H), so

∑_{k=0}^{k=N} a_{k}z^{k}→g.

(iii) ⟹ (i): Suppose Φ is in (N^{+}, τ)^{∗}, Φ(z^{k}) = α_{k}. If g ∼ ∑_{k=0}^{∞} a_{k}z^{k} is in N^{+},

Φ(g) = Φ( lim_{N→∞} ∑_{k=0}^{k=N} a_{k}z^{k})

= lim_{N→∞} ∑_{k=0}^{k=N} α_{k}a_{k}.

Thus ∑_{k=0}^{∞} α_{k}a_{k} converges for all g ∼ ∑_{k=0}^{∞} a_{k}z^{k} in N^{+}; so by the Banach-Steinhaus theorem [21, 9-3.7], Φ is in (N^{+}, H)^{∗}.

Conversely, another application of the Banach-Steinhaus theorem (using that τ is barrelled and Szegö) gives that (N^{+}, τ)^{∗} actually equals (N^{+}, H)^{∗}. Since both topologies are barrelled, hence Mackey, they must be the same. □

We remark that various other conditions can also be proved to be equivalent; for example, requiring that τ be metrizable and that the completion of (N^{+}, τ) be contained in the space of functions analytic on the open unit disk.

TOPOLOGIES ON THE SMIRNOV CLASS 9

3. Weighted Hardy Spaces

It is well-known that a weight w is in W if and only if there is an outer function h in H^{2} with |h|^{2} = w almost everywhere - see e.g [4] or [7]. Thus (N^{+}, H)^{∗} consists of those functionals in H^{2}(w)^{∗} for every w in W. If w is in W, any sequence of polynomials that converges in H^{2}(w) must converge almost uniformly on 𝔻, and hence converges to an analytic function on the disk (whose radial limits agree almost everywhere with the limit in H^{2}(w)). Thus evaluating Fourier coefficients extends continuously to H^{2}(w); knowing how rapidly the Fourier coefficients of a function in N^{+} can grow enables us to put a bound on the norms of these functionals. We believe that this “asymptotic Szegö theorem” is important, but it does not appear to have been noticed previously. Note that the bounds are for general w: for a specific w (e.g. w = 1), obviously the bounds can be improved.

Theorem 3.1. Let w be in W, and c > 0. Then there exists a constant C_{1} such that

sup ; ∫_{𝕋} |p|^{2}wdθ≤1 |p̂(n)| ≤ C_{1}e^{c\sqrt{n}}.

Moreover, e^{c\sqrt{n}} cannot be replaced by e^{c_{n}\sqrt{n}} for any sequence c_{n} converging to zero.

Proof. Let f_{n} be the functional in (N^{+}, H)^{∗} with f̂_{n}(k) = δ_{nk}e^{−c\sqrt{n}} (δ_{nk} is the Kronecker delta). These functionals are pointwise bounded, so they constitute an equicontinuous family, i.e. there is an open neighborhood of zero U in (N^{+}, H) such that | < g, f_{n} > | is less than one for all g in U and all f_{n}. For any w in W, U ∩ H^{2}(w) contains some open ball around zero, of radius ϵ say. Putting C_{1} = \frac{1}{ϵ}, we get that

sup ; ∫_{𝕋} |p|^{2}wdθ≤1 |p̂(n)| ≤ C_{1}e^{c\sqrt{n}}.

as desired.

That these bounds are best possible follows immediately from the existence, for each sequence c_{n} converging to zero, of a function f in some H^{2}(w) for which f̂(n) ≠ O(e^{c_{n}\sqrt{n}}).

□

For any element f in the dual of H^{2}(w) we will write f = ∑_{k=0}^{∞} f̂(k)z^{k} to mean < z^{k}, f >= \overline{f̂(k)}, so for any polynomial p = ∑_{k=0}^{N} a_{k}z^{k}, the action of f on p is given by

< p, f >= ∫_{𝕋} p\overline{f}dθ = ∑_{k=0}^{N} a_{k}\overline{f̂(k)}.

We say that g is a multiplier of H^{2}(w)^{∗} if, for every f in H^{2}(w)^{∗}, gf is in H^{2}(w)^{∗}. The following theorem says that the universal multipliers are precisely (N^{+}, H)^{∗}.

Theorem 3.2. The function g is a multiplier of H^{2}(w)^{∗} for every w in W if and only if g is in (N^{+}, H)^{∗}.

Proof. Necessity is clear, as 1 is in every H^{2}(w)^{∗}, so every multiplier must itself be in H^{2}(w)^{∗}.

For sufficiency, fix w in W, and f in H^{2}(w)^{∗}. Let g be given, with |ĝ(k)| ≤ C_{1}e^{−c\sqrt{k}} for some positive c. We must show fg is in H^{2}(w)^{∗}, i.e. there exists a constant K such that, for all polynomials p,

∗ = | ∫_{𝕋} p\overline{(gf)}dθ|^{2} ≤ K ∫_{𝕋} |p|^{2}wdθ.

Writing p = ∑_{k=0}^{N} a_{k}z^{k}, ∗ becomes

∗ = ∑_{j,k=0}^{n} a_{k}\overline{a}_{j}\overline{ĝf(k)}ĝf(j)

= ∑_{j,k=0}^{n} a_{k}\overline{a}_{j} ∑_{l=0}^{∞} \overline{ĝ(l)f̂(k − l)} ∑_{m=0}^{∞} ĝ(m)f̂(j − m)

= ∑_{l=0}^{∞} ∑_{m=0}^{∞} \overline{ĝ(l)}ĝ(m) ∑_{j,k=0}^{n} a_{k}\overline{a_{j}}\overline{f̂(k − l)}f̂(j − m)

where the interchange of summation signs is justified because {f̂(k − l)} is zero for l > k, so the sums are really finite.

Now,

∑_{k=0}^{N} a_{k}\overline{f̂(k − l)} = ∫_{𝕋} \overline{z}^{l}p(z)\overline{f}(z)dθ,

so if k(w, l, f) is defined by

k(w, l, f) = sup\{| ∫_{𝕋} \overline{z}^{l}p(z)\overline{f}(z)dθ| : p a polynomial, and ∫_{𝕋} |p|^{2}wdθ ≤ 1\},

then

∗ ≤ ∑_{l=0}^{∞} ∑_{m=0}^{∞} \overline{ĝ(l)}ĝ(m)k(w, l, f)k(w, m, f) ∫_{𝕋} |p|^{2}wdθ.

Write \overline{z}^{l}p(z) = F + P, where F is a polynomial in z, and P is a polynomial of degree less than or equal to l in \overline{z}, with no constant term. Then

k(w, l, f)^{2} = sup_{p} \frac{| ∫_{𝕋}(F + P)\overline{f}dθ|^{2}}{∫_{𝕋} |F + P |^{2}wdθ}

= sup_{p} \frac{| ∫_{𝕋} F \overline{f}dθ|^{2}}{∫_{𝕋} |F + P |^{2}wdθ} .

Because f is in H^{2}(w)^{∗}, there is a constant C_{2} such that

k(w, l, f)^{2} ≤ C_{2} sup_{p} \frac{∫_{𝕋} |F |^{2}wdθ}{∫_{𝕋} |F + P |^{2}wdθ} .

By lemma 3.3, this is less than or equal to C_{2}C_{3}e^{c\sqrt{l}}. Therefore

∑_{l=0}^{∞} ∑_{m=0}^{∞} \overline{ĝ(l)}ĝ(m)k(w, l, f)k(w, m, f) ≤ (C_{1})^{2}C_{2}C_{3} ∑_{l=0}^{∞} ∑_{m=0}^{∞} e^{−c\sqrt{l}}e^{−c\sqrt{m}}e^{\sqrt[\frac{c}{2}]{l}}e^{\sqrt[\frac{c}{2}]{m}}

and the right hand side sums to a finite value, C_{4} say. Therefore

∗ = | ∫_{𝕋} p\overline{(fg)}dθ|^{2} ≤ C_{4} ∫_{𝕋} |p|^{2}wdθ,

as desired. □

Lemma 3.3. Let w be in W, l be a positive integer, and let c > 0 be given. Then there is a constant C_{3}, depending only on c and w, such that for any analytic polynomial F, and any co-analytic polynomial P of degree l and with no constant term,

\frac{∫_{𝕋} |F |^{2}wdθ}{∫_{𝕋} |F + P |^{2}wdθ} ≤ C_{3}e^{c\sqrt{l}}.

12 JOHN E. MCCARTHY

Proof. Without loss of generality, we can assume that ∫_{𝕋} |F |^{2}wdθ = 1, and that ∫_{𝕋} |F + P |^{2}wdθ ≤ \frac{1}{4}, and ∫_{𝕋} |P |^{2}wdθ ≥ \frac{1}{4}. Write P (z) = ∑_{k=1}^{l} α_{k}\overline{z}^{k}. Then

∫_{𝕋} |P |^{2}wdθ = ∑_{j,k=1}^{l} α_{k}\overline{α}_{j} ∫_{𝕋} z^{k−j}wdθ

≤ ∑_{j,k=1}^{l} |α_{k}||α_{j}|∥w∥_{1}.

So there exists some m between 1 and l for which

|α_{m}| ≥ \frac{∥P ∥_{H^{2}(w)}}{l\sqrt{∥w∥_{1}}}

≥ \frac{1}{2l∥w∥_{1}} .

But ∫_{𝕋} |F + P |^{2}wdθ = ∫_{𝕋} |z^{l}(F + P)|^{2}wdθ, and z^{l}(F + P) is an analytic polynomial with one of its first l coefficients greater than or equal to \frac{1}{2l∥w∥_{1}} , so by Theorem 3.1 there is a constant C_{1} such that ∫_{𝕋} |z^{l}(F + P)|^{2}wdθ ≥ C_{1}e^{−c\sqrt{l}}. This gives the desired conclusion.

□

We remark that in [10], Helson and Szegö found necessary and sufficient conditions for a weight w in W to satisfy

sup_{F,P} \frac{∫_{𝕋} |F |^{2}wdθ}{∫_{𝕋} |F + P |^{2}wdθ} < ∞

where F is an analytic polynomial and P is a co-analytic polynomial with no constant term. The condition is that log(w) be writable as u + ṽ, where u and v are in L^{∞}(𝕋, θ), ∥v∥_{∞} < \frac{π}{2} , and ṽ is the harmonic conjugate of v. For weights which do not satisfy the Helson-Szegö condition, Lemma 3.3 can be thought of as an asymptotic version.

4. Conclusion

Given a uniform algebra, with a distinguished homomorphism with (say) a unique representing measure, one can define a Smirnov class, and most of the classical theorems go through - see [6], where it is called the Hardy algebra. In particular, the Smirnov class

TOPOLOGIES ON THE SMIRNOV CLASS 13

can be represented as a union of Hilbert spaces [6, V.4.4], and one can therefore define an inductive limit topology as above. How far can the results of this paper be extended to the abstract setting? For example, if the kernel of the homomorphism is singly generated by some function F, when can one represent the Smirnov class as convergent power series in F?

What is the Bass stable rank of N^{+}? It is easy to see that the topological stable rank of (N^{+}, H) is 2, but because the invertible elements are not open, there is no a priori reason that this should dominate the Bass stable rank. Determining the Bass stable rank of H^{∞} is hard - see [13], where a lot of progress has been made - but, because all outer functions are invertible in N^{+}, the problem here might be somewhat easier.

A final remark, for topological vector space enthusiasts: (N^{+}, H) is a natural example of an ultrabornological space (i.e. the inductive limit of Banach spaces) that is not quasicomplete (for this, plus the fact that the dual is reflexive, would force (N^{+}, H) itself to be reflexive - [11, p. 303]).

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14 JOHN E. MCCARTHY

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TOPOLOGIES ON THE SMIRNOV CLASS 15

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Department of Mathematics Indiana University Bloomington IN 47405 USA