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

In the scalar case, the spectral factorization mapping f→f+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\rightarrow f^+$$\end{document} puts a nonnegative integrable function f having an integrable logarithm in correspondence with an outer analytic function f+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^+$$\end{document} such that f=|f+|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f = |f^+|^2$$\end{document} is almost everywhere. The main question addressed here is to what extent ‖f+-g+‖H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f^+ - g^+\Vert _{H_2}$$\end{document} is controlled by ‖f-g‖L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f-g\Vert _{L_1}$$\end{document} and ‖logf-logg‖L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \log f - \log g\Vert _{L_1}$$\end{document}.


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 log f ∈ L 1 (T). (1) Then it admits a spectral factorization 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 Eq. (2) holds for these boundary values. Note also that f + = f − a.e. on T and therefore (2) is equivalent to Condition (1) is necessary for factorization (2) to exist. It also plays an important role in the linear prediction theory of stationary stochastic processes, one of the historically first applications of spectral factorization (see [16,21]). Namely, let . . . , X −1 , X 0 , X 1 , . . . be a stationary stochastic process with the spectral measure dμ = f dt + dμ s . In a different but equivalent language, {X n } n∈Z is a sequence in a Hilbert space and X n , X k = 1 2π T e i(n−k)θ dμ(θ). The process is deterministic if X n+1 can be represented as the limit of linear combinations of vectors from {. . . , X n−1 , X n }, i.e, X n+1 ∈ Span{. . . , X n−1 , X n }. As it happens (see e.g., [16]), condition (1) is necessary and sufficient for the process to be non-deterministic.
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 f → f + (4) 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, A proof of an analog 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

Theorem 1
There exist functions f n , g n ≥ 0, n ∈ N, such that Nevertheless, one can still obtain an estimate for g + − f + H 2 if one takes into account that for each f ∈ L 1 (T) there exists an Orlicz space L (T) such that f ∈ L (T) (see, e.g., [17,Sect. 8]). One can show that there exists a function : [0, +∞) → [0, +∞) such that lim t→0 (t) = 0 and (see Theorem 3 below). The estimate becomes particularly simple if f ∈ L ∞ (T).
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 toward the investigation of similar problems in the more complicated matrix case, which is going to be the subject of a forthcoming paper.
Using Kolmogorov's weak type (1, 1) estimate with constant K , one gets We need some notation from the theory of Orlicz spaces (see [17,18]). Let and be mutually complementary N -functions, i.e.,  Let ( , , μ) be a measure space, and let L ( ), L ( ) be the corresponding Orlicz spaces, i.e., L ( ) is the set of measurable functions on for which either of the following norms is finite. Note that these two norms are equivalent, namely (see, e.g., [17, (9.24)] or [18,Sect. 3.3,(14)]) We will use the following Hölder inequality (see, e.g., [17, (9.27)] or [18, Sect. 3.3, For an N -function , let If is continuous, the above definition of can be rewritten in terms of inverse functions, because is nondecreasing. For an arbitrary N -function , one has Hence, It is clear that (s) → 0 as s → 0 + . Also, Lemma 2 For every N -function , the following estimate holds where K 0 := K 2 π 0 sin λ λ dλ < 1.25 (13) and K is the same as in Lemma 1.
Proof We will use Lemma 1 with and a = π . We have Taking κ > 2 K 0 ψ L 1 , we observe that the right-hand side of inequality (14) does not exceed 1 by virtue of the definition (9). Thus, (12) follows.
where K 0 is defined by (13).

Theorem 3
For every pair and of mutually complementary N -functions, the following estimate holds where K 0 is the same as in Lemma 2. Proof Here and below, for x ∈ R, we define x + := max(x, 0). Using the Hölder inequality (8) and the elementary inequality which is easily proved by considering the cases a ≥ b and a < b separately, one gets It is now left to apply Lemma 2 with ψ = 1 2 (log f − log g).
Since every integrable function belongs to a certain Orlicz space (see [17,Sect. 8]), Theorem 3 with an appropriate pair and of mutually complementary N -functions applies to any nonnegative integrable function f with an integrable logarithm. The condition (10) is fulfilled as well.
such that w n (0) = −ε n /2, where ε n = 1 2π n . Let h n := exp(Re w n ). Then (log h n ) ∼ = (Re w n ) ∼ = Im w n and therefore 1 − cos( 1 2 (log h n ) ∼ ) L ∞ = 2. Due to duality considerations, there exists f 0 n ≥ 0 such that f 0 n L 1 = 1 and If log f 0 n ∈ L 1 , we take f n = f 0 n . Otherwise, we define f n = (1−ε n /2) f 0 n +ε n /4π . Then, f n L 1 = 1, and Finally, let g n = h n f n . Then, 0 ≤ g n ≤ f n , g n L 1 ≤ 1, 2n , log f n − log g n L 1 = log h n L 1 ≤ 2π log h n L ∞ = 2πε n = 1 n , and f + n − g + n 2 H 2 ≥ 2(1 − ε n ) 1 − cos Remark The norms log f n L 1 and log g n L 1 might not be bounded in Theorem 1. Let f 0 n be the function from the above proof. Changing the definition of f n in the proof to f n = f 0 n + 1, one can change the estimates f n L 1 , g n L 1 ≤ 1 in the theorem for f n L 1 = 2π + 1, g n L 1 ≤ 2π + 1, log f n L 1 ≤ 1.
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