A COMPOSITION PROBLEM INVOLVING ANALYTIC FUNCTIONS

Paul E. Fishback

1. Introduction. One of the beautiful principles of Fourier analysis maintains that when an analytic function ϕ acts upon a function f continuous on the unit circle T in the complex plane, the composition ϕ(f) frequently inherits “nice” properties possessed by f. An example of this idea is the Wiener–Levy Theorem, which asserts that if f has an absolutely convergent Fourier series and if ϕ is a function analytic in a neighborhood of f(T), then the composition ϕ(f) has an absolutely convergent Fourier series as well [4].

In this note we consider a composition problem in which the “nice” property can be expressed in terms of analyticity or vanishing Fourier coefficients. Specifically, we let A(D) (sometimes referred to as the “disk algebra”) denote the algebra of functions that are continuous on T and that possess analytic extensions into the open unit disk D. Equivalently, owing to the Poisson integral, a function f belongs to A(D) if and only if it is continuous on T and has the property that its Fourier coefficients

f̂(n) = \frac{1}{2π} ∫_{0}^{2π} f(t)e^{−int}dt

are equal to zero for all n < 0.

Clearly if f belongs to A(D) and if φ is analytic in a neighborhood of f(\overline{D}), then g = φ(f) belongs to A(D), and one may express its extension in D as g(z) = φ(f(z)). Moreover, the elementary examples consisting of the pairs φ(z) = z^{2}, f(z) = \sqrt{z}, and φ(z) = e^{z}, f(z) = log(z) (where any branch of f is chosen) illustrate how an analytic function φ can map a function f not in A(D) to a function g = φ(f) back in this class.

Yet for neither of these two examples is the function f continuous on the unit circle. A more difficult question, one raised by Forelli [2], arises when one asks whether, for such an arbitrary function φ, there exists a function f continuous on T and not in A(D) that is mapped to a function g = φ(f) in A(D). In terms of analyticity, g = φ(f) has an analytic extension g(z) in D even though f, while continuous on T, does not possess such an extension. In terms of Fourier coefficients, f and g are continuous functions with the properties ĝ(n) = { \atop φ(f)}(n) = 0 for all n < 0 yet f̂(n) ≠ 0 for at least one n < 0. We show that the answer to this question depends upon the choice of φ. Interestingly, the answer differs for the two choices φ(z) = z^{2} and φ(z) = e^{z}.

2. Results. We begin with a theorem that uses classical complex analysis to provide the answer for a class of functions that includes φ(z) = z^{2}.

\underline{Theorem 1}. Let φ(z) be analytic in some region X, where its derivative has at least one zero. Then there exists a function f that is continuous on T, that does not belong to A(D), that satisfies f(T) ⊆ X, and that is mapped by φ to a function g = φ(f) in A(D).

\underline{Proof}. Suppose that φ′(α) = 0 and φ(α) = r. By considering the function φ(z+α)−r, we see that it suffices to prove the theorem for the case when X contains the origin and φ has a zero of order k ≥ 2 at the origin. For |w| less than some fixed positive δ, a classical result sometimes known as Weierstrass’ Vorbereitungssatz [3], states that the equation φ(z) = w has k roots z_{1}(w), …, z_{k}(w) of the form z_{i}(w) = g(w^{1/k}), where g(ζ) is a function analytic in some neighborhood of the origin and where the k-determinations of w^{1/k} yield the different branches of z_{1}(w), …, z_{k}(w). In particular, each z_{j}(w) is analytic in D(0; δ)\(−∞, 0] and is continuous and equal to 0 at w = 0.

Now choose λ > 0 with the property that 2λ^{2} < δ. Under the map w = λ^{2}(ζ^{2} + 1) the disc |ζ| < 1 is mapped to the disc |w − λ^{2}| < λ^{2} so that the functions z_{j}(λ^{2}(ζ^{2} + 1)) belong to A(D) and are all equal to 0 at the points ±i. Let

f(e^{it}) = \{ \begin{matrix}z_{1}(λ^{2}(e^{2it} + 1)), & if ℝ(e^{it}) ≥ 0 \\ z_{2}(λ^{2}(e^{2it} + 1)), & if ℝ(e^{it}) < 0. \end{matrix}

Then f is continuous on T, has its range contained in X, and is mapped by φ to

φ(f(e^{it})) = φ(z_{j}(λ^{2}(e^{2it} + 1))) = λ^{2}(e^{2it} + 1) ∈ A(D).

However, if f itself belongs to A(D), then so do the differences f −z_{1}(λ^{2}(e^{2it} + 1)) and f −z_{2}(λ^{2}(e^{2it} +1)). Since the nullset on T of any nonzero function in A(D) has Lebesgue measure zero [4], it follows that z_{1} ≡ z_{2} in D(0; δ)\(−∞, 0], a fact that contradicts the manner in which these roots were chosen. Thus, f itself does not belong to A(D).

Ideally, proving that a function φ with nonvanishing derivative is capable of mapping a continuous function f to g = φ(f) in A(D) only if f itself belongs to A(D) would provide a complete answer to Forelli’s question. The pair φ(z) = \frac{1}{z}, f(e^{it}) = e^{−it} illustrates, however, that this is not the case.

Yet, this example distinguishes itself in that there exists no simply connected region containing f(T) yet not containing zero, the exceptional value of φ. The following theorem illustrates that it is precisely this fact that prevents f from belonging to A(D).

We state this theorem using a generalization of the idea of exceptional value, known as the asymptotic value. The function φ analytic in the region X is said to have an asymptotic value at the complex number α if there exists a continuous mapping z from [0, 1) into X with the properties that φ(z(t)) → α and z(t) → B(X), the ideal boundary of X in the extended complex plane, as t → 1. It is a well–known fact that if an analytic function φ has a nonvanishing derivative, the only possible singularities of a branch of its inverse occur at asymptotic values [1].

\underline{Theorem 2}. Suppose φ is analytic in some region X where its derivative is nonvanishing. Let f be a continuous function on T for which f(T) ⊆ X and g = φ(f) belongs to A(D). If there exists a simply connected region Ω satisfying f(T) ⊆ Ω ⊆ X and having the property that φ(Ω) contains no asymptotic values of φ, then f itself belongs to A(D).

\underline{Proof}. We begin with the claim g(\overline{D}) ⊆ φ(Ω). If this inclusion does not hold, then there exists some β in g(D) with the property that φ − β is zero–free on Ω. Hence, φ − β possesses an analytic logarithm G on Ω. If we let Ind_{g}(β) denote the usual winding number of the curve g(T) with respect to β and if we note that Ω is simply connected, we conclude that

Ind_{g}(β) = Ind_{φ(f)}(β) = Ind_{e^{G(f)}}(0) = 0.

But the Argument Principle dictates that Ind_{g}(β) also represents the number of zeros in D of g − β. From this contradiction, the claim follows.

Proceeding with the proof, we choose a function element ψ analytic in a neighborhood of g(1) with the property that ψ(g(1)) = f(1). The element ψ(g) then analytically continues along all arcs in D that start at z = 1, since g(\overline{D}) contains no asymptotic values of φ. By the Monodromy Theorem, ψ(g) defines an analytic function in D whose extension to T agrees with f. Hence, f belongs to A(D).

3. Discussion. The hypothesis of the preceding theorem is always satisfied if the function φ is analytic in some simply connected region X, has a nonvanishing derivative there, and possesses no asymptotic values. The simplest example of such a function is of course φ(z) = e^{z}. Another example includes the so–called modular function [5].

While these results answer Forelli’s question in large part, they do leave situations unaccounted for. Such situations occur when one assumes in the preceding hypothesis that X is simply connected and drops altogether the condition regarding asymptotic values. An interesting example of a function φ that leads to such a situation, one for which the author does not know the outcome to the original question, is φ(z) = ∫_{w=0}^{w=z} e^{w^{2}}dw.

\underline{References}

1. L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1960.

2. F. Forelli, University of Wisconsin–Madison, personal correspondence, 1990.

3. E. Hille, Analytic Function Theory, Vol. I, 2nd ed., Chelsea Publishing Company, New York, 1962.

4. Y. Katznelson, An Introduction to Harmonic Analysis, 2nd ed., Dover Publications, New York, 1976.

5. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw–Hill Book Company, New York, 1987.

Paul E. Fishback Department of Mathematics and Statistics Grand Valley State University Allendale, MI 49401 email: fishbacp@gvsu.edu