Quantitative results on continuity of the spectral factorization mapping in the scalar case

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $\|\log f - \log g\|_{L_1}$.


Introduction
Let f be a nonnegative integrable function on the unit circle in the complex plain, 0 ≤ f ∈ L 1 (T), satisfying the Paley-Wiener condition (1) log f ∈ L 1 (T).
Then it admits a spectral factorization (2) f (t) = f + (t)f − (t) a.e. on T, where f + is a function analytic inside the unit circle, f + ∈ A(T + ), and f − (z) = f + (1/z), which is analytic outside the unit circle including the infinity, f − ∈ A(T − ). More specifically, f + belongs to the Hardy space H 2 (D), therefore its boundary values f + (t) = f + (e iθ ) = lim r→1 f + (re iθ ) exist a.e. and the equation (2) holds for these boundary values. Note also that f + = f − a.e. on T and therefore (2) is equivalent to f (t) = |f + (t)| 2 a.e. on T.
If we require f + to be an outer analytic function, then the factorization (2) is unique up to a constant factor c with absolute value 1, |c| = 1. The unique spectral factor which is positive at the origin can be a priori written as In most applications, a spectral factor f + in (2) is not explicitly required to be outer and instead is subject to certain extremal conditions called, in various works, minimal phase or maximal energy, optimal, etc. In mathematical terms, however, they amount to f + being outer, so seeking the solution (3) is natural. From the practical point of view, it is important to study the continuity properties of the spectral factorization map (4) f → f + defined by (3). Namely, we are interested in knowing how close g + is to f + when a spectral density g is close to f . The reason why we study this question is that usually an estimated spectral density functionf being dealt with is constructed empirically from observations and is only an approximation to the theoretically existing spectral density f . Therefore we need to know how closef + remains to f + under such approximation. An answer to the above question depends on norms we use as a measurement of the accuracy in the spaces of functions and of their spectral factors. Since the boundary values of the function (3) can be expressed as where ∼ stands for the harmonic conjugation operator and the conjugation is not a bounded operator on L ∞ or C(T), it is not surprising that the map (4) is not continuous in these spaces [1]. Furthermore, it is shown in [5] that every continuous function on T is a discontinuity point of the spectral factorization mapping in the uniform norm, whereas in [14] it was shown that on a large class of function spaces (the so called decomposing Banach algebras) the spectral factorization mapping is continuous. The spectral factorization of a trigonometric polynomial which is non-negative on T, has the form i.e. the spectral factor f + is a polynomial of the same degree N. This result is known as the Fejér-Riesz lemma (see, e.g. [8]). The spectral factor can also be expressed in terms of zeros of polynomial (5), and therefore the map (4) is continuous on P N , the set of all functions of the form (5). Papers [6], [7] are devoted to estimating the constant C N in the inequality and it is shown there that C N ∼ log N asymptotically, under the condition that the values of functions φ and ψ are bounded away from 0.
Moving to Lebesgue spaces, the map (4) is not continuous in the L 1 norm in general, since a small change of values of function f , if these values are close to 0, may cause a significant change of log f . However, (6) f A proof of an analogue of (6) for more general matrix case can be found in [3] or [10]. In the present paper, we discuss quantitative estimates of the rate in the above convergence. Firstly, we look for estimates of g + − f + H 2 in terms of g − f L 1 and log g − log f L 1 . It turns out that, in general, there is no such estimate. Namely, there is no function Π : [0, +∞) 2 → [0, +∞) such that lim s,t→0 Π(s, t) = 0 for which the estimate
Theorem 2. Let f and g be arbitrary spectral densities for which f + and g + exist. Then The paper is organized as follows. In Sect. 2, we prove (7) and Theorem 2. Theorem 1 is proved in Sect. 3. This paper is a preliminary step towards the investigation of similar problems in the more complicated matrix case, which is going to be the subject of a forthcoming paper.

Positive Results
Let K be the best constant in Kolmogorov's weak type (1, 1) inequality where m stands for the Lebesgue measure on the real line. It is known that .347 (see [9]).
For an N-function Φ, let If Φ ′ is continuous, the above definition of Λ Φ can be rewritten in terms of inverse functions, because Φ ′ is nondecreasing. For an arbitrary N-function Φ, one has It is clear that

Lemma 2.
For every N-function Φ, the following estimate holds and K is the same as in Lemma 1.
where K 0 is defined by (13).
Proof. We will use Lemma 1 with where K 0 is the same as in Lemma 2.