Approximation problems in some holomorphic spaces, with applications

I. Chalendar, J. Leblond, and J. R. Partington

Abstract. We review a selection of extremal problems to do with constrained approximation in certain Banach spaces of holomorphic functions, including the classical Hardy spaces and Paley–Wiener spaces. In many cases the solutions are best interpreted in terms of linear operators. Applications of the problems under discussion to systems identification, signal processing, inverse problems for partial differential equations, and operator theory are presented.

1. Introduction

In recent years much research effort has been focused on certain approximation and recovery problems in classes of analytic functions. These include the following classes of problem, which we review in this paper because of their applications in other fields of analysis. The following preliminary descriptions are necessarily brief and fuller details follow later.

1. Approximation: given a function defined on some set, find a holomorphic function which approximates it on this set. Further constraints may be imposed upon the approximating function.

2. Extension: given a function defined on some set, extend its definition to a larger set so that the extended function is close to a holomorphic function.

3. Recovery: given the values of a holomorphic function on a subset of its domain of definition, recover the function.

4. Correction: given an unbounded holomorphic function, approximate it by a bounded function.

The spaces of holomorphic functions which principally concern us are the Hardy spaces (either on a disc or on a multiply-connected domain) and the Paley–Wiener spaces. These spaces arise naturally in a variety of applications, of which we discuss the identification of linear systems, signal processing, inverse problems for partial differential equations, and factorization results for linear operators. The solutions to these problems are operator-theoretic in nature, and we make use of Hankel and Toeplitz operators and their variants.

The layout of this paper is as follows. Section 2 treats various extremal problems, mostly in the context of Hilbert function spaces. Section 3 analyses the Toeplitzlike operators which appear in the solution of many such problems, with an emphasis on spectral theory. The applications to systems identification are described in Section 4. We continue by describing some variants of the extremal problems, where complex functions are constrained by their real or imaginary parts, in Section 5. More applications follow: to signal processing in Section 6, and to inverse problems for partial differential equations in Section 7. Finally, in Section 8, we discuss further approximation results in Hardy spaces, and their applications, which include abstract spectral theory of linear operators.

2. Bounded extremal problems

2.1. Bounded extremal problems in Hardy classes The systematic approach to extremal problems in Hardy spaces H^{p} of the disc goes back to [48] and [68] in around 1950, and has been extensively developed since then, see [36, chap. 8], [43, chap. IV], [59] and references therein. For integers p, q, with 1 ≤ p ≤ ∞ and 1/p+1/q = 1, given a function f ∈ L^{p}(𝕋), these extremal problems concern both sides of the following duality relation (a consequence of the Hahn–Banach theorem):

min_{g∈H^{p}} ∥f − g∥_{L^{p}(𝕋)} = sup_{∥ϕ∥_{L^{q} (𝕋)}≤1^{ϕ∈H{q \atop 0}}} \frac{1}{2π} ∫_{𝕋} f(z) ϕ(z) dz .  (1)

The above left-hand side minimization issue is usually called the dual extremal problem associated to f and it is known that:

• it always admits a solution,

• for 1 ≤ p < ∞, the solution is unique (computational aspects are clear for p = 2 but not in general, see [19, 36]),

• for p = ∞ (Nehari extension), the solution is unique and the error has constant modulus if f ∈ H^{∞} + C(𝕋). In this case, there is an associated Hankel operator Γ_{f} which is compact and singular value decomposition provides a way to get the solution (although this is effective only for rational f and some care has to be taken with continuity properties of best approximation operators [80]). This is still true for the problem of best uniform meromorphic approximation, known as the Adamjan–Arov–Krein (AAK, for short) problem, see [1] and Section 4.1.

The Hankel operator mentioned above, with symbol f ∈ L^{∞}(𝕋), can be defined by the equation

Γ_{f}u = P\overline{H}_{0}^{2}(fu),  u ∈ H^{2},  (2)

Approximation problems in some holomorphic spaces, with applications 3

where P\overline{H}_{0}^{2} : L^{2}(𝕋) → \overline{H}_{0}^{2} denotes the orthogonal projection (see, for example, [75, 92]). Let us write 𝕋 = I ∪ J, the union of two disjoint arcs both of positive Lebesgue measure, and assume that we are given a function f ∈ L^{p}(I) that we want to approximate by traces on I of H^{p} functions. For such an issue to be wellposed, we must add some constraint concerning the behaviour of the approximants on the complementary part J (this follows from density properties of H_{|I}^{p} in L^{p}(I), see [13, prop. 1], [11, prop. 3]). For Ψ ∈ L^{p}(J), and M a positive real number, we define the subset B_{M,Ψ}^{p} of H^{p} by:

B_{M,Ψ}^{p} := {g ∈ H^{p} : ∥Ψ − g∥_{L^{p}(J)} ≤ M} .

With these notations, the Bounded Extremal Problem we consider can be stated as follows.

Problem 2.1. For f, Ψ, M as above, find g_{p} ∈ B_{M,Ψ}^{p} such that

∥f − g_{p}∥_{L^{p}(I)} = min_{g∈B_{M,Ψ}^{p}} ∥f − g∥_{L^{p}(I)} := β_{p} .

Instances of such a problem for p = 2 have been treated in [64] with f = 0 and in [5] for Ψ = 0. We next review some properties of Problem 2.1 which have been established in [11, 13] and which generalize the ones related to dual extremal problems on 𝕋 and recalled above. Note also that band-limited versions of Nevanlinna–Pick or Carathéodory–Fejér interpolation issues can also be rephrased as Problem 2.1: see the unpublished report [14]. Although a duality relation is no longer available on I for the class B_{M,Ψ}^{p} of approximants, it can be replaced by uniform convexity properties for 1 < p < ∞, by combining differentiation with more refined results on Douglas algebras for p = 1, and by rephrasing the problem as a weighted one on the whole of 𝕋 when p = ∞. Put C_{M,Ψ}^{p} := {g_{|I} : g ∈ B_{M,Ψ}^{p} }. Here, and later, the notation p∨q denotes a function equal to p on I and to q on J.

Theorem 2.2. (i) For 1 ≤ p < ∞, there exists a unique solution g_{p} to Problem 2.1; moreover, ∥Ψ − g∥_{L^{p}(J)} = M whenever f ∉ C_{M,Ψ}^{p}. (ii) For p = ∞, if f ∨Ψ ∈ H^{∞} +C(𝕋) , then there exists a solution g_{∞} to Problem 2.1; if moreover f ∉ C_{M,Ψ}^{∞}, then g_{∞} is unique and f − g_{∞} and Ψ − g_{∞} have constant modulus β_{∞} and M a.e. on I and J respectively.

Related constructive aspects and computational issues are described in Section 3.1 in the Hilbert case p = 2 which is our main concern in the remainder of this work except, however, for Section 4 where approximation issues in the uniform norm are again considered.

2.2. Bounded extremal problems in Hilbert spaces Suppose that we are given a Hilbert space H that can be decomposed into two distinct orthogonal direct sums of subspaces as:

H = H_{+} ⊕ H_{−} = K_{+} ⊕ K_{−}.  (3)

Given a function in K_{+}, we want to approximate it as well as possible by the projection on K_{+} of an element of H_{+} whose projection on K_{−} has controlled norm. Two decompositions that we have in mind here (and we discuss others later) are the Hardy space decomposition discussed in Section 3.1:

L^{2}(𝕋) = H^{2} ⊕ \overline{H}_{0}^{2} = L^{2}(I) ⊕ L^{2}(J),  (4)

with J = 𝕋 \ I, and the Paley–Wiener space decomposition discussed in Section 6:

L^{2}(ℝ) = P W (Ω) ⊕ P W (ℝ \ Ω) = L^{2}(J) ⊕ L^{2}(ℝ \ J),

In most applications we think of H_{+} as a class of holomorphic functions, and P_{K_{+}} as a restriction mapping. We shall discuss only the case when decomposition (3) holds with K_{+} ∩ H_{−} = {0}, when it follows easily that P_{K_{+}}H_{+} is dense in K_{+}.

Problem 2.3. Let M > 0 be a constant, and let k_{2} ∈ K_{−} be such that

S_{M} := {h ∈ H_{+} : ∥P_{K_{−}}h − k_{2}∥_{H} ≤ M} ≠ ∅.

For k_{1} ∈ K_{+}, with k_{1} ∉ P_{K_{+}}S_{M} find h_{0} ∈ S_{M} such that

∥P_{K_{+}}h_{0} − k_{1}∥_{H} = min{∥P_{K_{+}}h − k_{1}∥_{H} : h ∈ S_{M}} =: β, say.

Such a problem has been handled in [66] and provides a suitable framework in order to unify the formulation of various approximation issues that involve restrictions of analytic functions to subsets of the boundary of their domain.

Theorem 2.4. [66] Problem 2.3 has a unique solution h_{0}, the constraint is saturated (i.e., ∥P_{K_{−}}h_{0} − k_{2}∥_{H} = M), and the solution is

h_{0} = (I + λT)^{−1}P_{H_{+}}(k_{1} + (1 + λ)k_{2}),

where T : H_{+} → H_{+} is the self-adjoint operator P_{H_{+}}P_{K_{−}}, and λ > −1 is a constant such that ∥P_{K_{−}}h_{0} − k_{2}∥_{H} = M.

Companion to the above is an Extension Problem (or Bounded Completion Problem), which can be seen as one of extending the definition of a partly-specified function in order that the extended function be as close to a holomorphic function as possible.

Problem 2.5. Given k_{2} ∈ K_{−} and M > 0, let R_{M} = {k ∈ K_{−} : ∥k − k_{2}∥_{H} ≤ M}. Given k_{1} ∈ K_{+}, find k_{0} ∈ R_{M} such that

dist(k_{1} + k_{0}, H_{+}) = min_{k∈R_{M}} dist(k_{1} + k, H_{+}).

This problem is analysed in [66] in the general Hilbertian framework, having been treated for H^{2} in [14, 18]. The analogous H^{∞} problem has also been solved [15]. In any of these situations the appropriate solution can be constructively deduced from that of the related Bounded Extremal Problem 2.3. We mention also an extrapolation result from [66], which generalizes a theorem of Patil (see [78, 79]), on the recovery of functions in H^{2} from their values on a subset of the unit circle. The approximants produced by the next result coincide with those constructed by Patil, which are discussed in fuller detail in Section 3.2.

Theorem 2.6. [66] Given a decomposition (3) with K_{+} ∩H_{−} = {0} and K_{−} ∩H_{+} = {0}, and given k_{1} ∈ K_{+}, let

f_{λ} = (I + λT)^{−1}P_{H_{+}}k_{1}.

Then k_{1} ∈ P_{K_{+}}H_{+} if and only if (f_{λ}) remains bounded as λ → −1, in which case (f_{λ}) tends in norm to the unique h ∈ H_{+} such that P_{K_{+}}h = k_{1}.

Constructive aspects of Problem 2.3 have been approached in a number of particular instances of the decomposition (3). For Decomposition (4), P_{K_{+}}g = χ_{I}g for all g ∈ H, and ∥f − g∥_{L^{2}(I)} = ∥χ_{I}(f − g)∥_{L^{2}(𝕋)}; the problem coincides with Problem 2.1 for p = 2 and is discussed further in Section 3.1. Hardy spaces for circular domains also come up in this set-up, see Section 3.3, as well as the Paley–Wiener classes already referred to, see Section 6.

Decomposition (3) may also be considered in more general Banach spaces. For example, one can write

L^{p}(𝕋) = H^{p} ⊕ \overline{H}_{0}^{p} = L^{p}(I) ⊕ L^{p}(J), 1 < p < ∞ .

(see [53, chap. 9]) and thus return to Problem 2.1. Another potentially interesting decomposition involves the notion of a Decomposing Banach Algebra [33, 56], of which the Wiener algebra of all absolutely convergent Fourier series is the most familiar example. However, little general analysis has been done in the non-Hilbertian case.

3. Toeplitz operators and spectra

3.1. Hardy classes in the disc Without loss of generality we take I = (e^{−ia}, e^{ia}) for some a with 0 < a < π. We work with the particular form (4) of decomposition (3):

L^{2}(𝕋) = H^{2} ⊕ \overline{H}_{0}^{2} = L^{2}(I) ⊕ L^{2}(J),

and we present some analysis given in [9]. In this case, T is the Toeplitz operator T : H^{2} → H^{2} with symbol χ_{J}. Following the constructive diagonalization procedure for self-adjoint Toeplitz operators given in [84], we find that T is unitarily equivalent to multiplication M_{x} by the independent variable on L^{2}([0, 1], dρ), where dρ(x) = π^{−1}(sin a)dx. Indeed, M_{x} = V T V^{−1}, where V : H^{2} → L^{2}([0, 1], dρ) is the unitary transformation whose effect on Cauchy kernels k_{α}(z) = 1/(1 − \overline{α}z) is given by

(V k_{α})(x) = [\overline{ψ_{x}(α)}(1 − \overline{α}e^{ia})^{1/2}(1 − \overline{α}e^{−ia})^{1/2}]^{−1},

and where, for 0 < x < 1, ψ_{x} is the unique outer function such that ψ_{x}(0) > 0 and such that |ψ_{x}|^{2} is equal to x on I and to 1 − x on J.

Calculations using the transformation V enable one to compare the quantities e = e(λ) := β_{2} and M = M(λ), in the following version of the Bounded Extremal Problem 2.1 with p = 2, which also coincides with Problem 2.3 for the decomposition (4).

Problem 3.1. Given functions f ∈ L^{2}(I), Ψ ∈ L^{2}(J), and a positive constant M, find g_{2} ∈ B_{M,Ψ}^{2} such that

e := ∥f − g_{2}∥_{L^{2}(I)} = min_{g∈B_{M,Ψ}^{2}} ∥f − g∥_{L^{2}(I)} .

Here λ gives an implicit parametrization of the optimal solutions g_{2}; the error estimates satisfy e(λ) → 0 and, if f ∉ H_{|I}^{2}, M(λ) → ∞ as λ → −1. Note that the optimal g_{2} = g_{λ} is given by

g_{λ} = (1 + λT)^{−1}P_{H^{2}}(f ∨ (1 + λ)Ψ),  (5)

for the unique λ > −1 such that ∥Ψ − g_{λ}∥_{L^{2}(J)} = M when f ∉ C_{M,Ψ}^{2} , and for λ = −1 if f ∈ C_{M,Ψ}^{2} . A weighted version of Problem 3.1 has been considered in [65] while a smoother one, namely in Hardy–Sobolev spaces, has been treated in [10]. The following result from [9] is a sample of what can be proven by translating the expressions for the approximation errors into integrals and then estimating them. Earlier results in this direction can be found in [5, 11].

Theorem 3.2 ([9]). Suppose that f satisfies

(1 − e^{−iθ}e^{ia})^{−1/2}(1 − e^{−iθ}e^{−ia})^{−1/2} f(e^{iθ}) ∈ L^{1}(I) ,

and

(1 − e^{−iθ}e^{ia})^{1/2}(1 − e^{−iθ}e^{−ia})^{1/2} f(e^{iθ}) ∈ W^{1,1}(I)

(the Sobolev class consisting of functions with derivatives in L^{1}(I)). Then

M(λ)^{2} = o((λ + 1)^{−1} log^{−2}(λ + 1)),

as λ → −1, while

e(λ)^{2} = o(| log^{−1}(1 + λ)|).

3.2. Recovery problems Faster convergence rates are available if f is already the restriction of an H^{2} function (to be denoted again by f). In that case, at least if M is large enough, Problem 3.1 amounts to reconstructing f ∈ H^{2} from the knowledge of f_{|I}. We touched upon these ideas in Section 2.2, and now we explore them further.

Recovery issues for analytic functions from traces on the boundary were originally approached by Carleman, Goluzin and Krylov (see [3, 79] and the bibliography therein). It appears that recovery procedures of the Carleman type are strongly related to the above approximation scheme to f ∈ L^{2}(I) and that the boundedness on J of such approximants provides criteria to decide whether f is the trace of some H^{2} function or not, a so–called Loewner type question that was taken up in [3, 6, 64, 67, 79, 83], for instance.

If f ∈ H_{|I}^{2} and Ψ = 0, then the solution g_{λ} given by (5) coincides with the Patil’s sequence of approximants (f_{α}) defined below by (7) and built for recovery purposes rather than approximation, [11, 78, 79] . Put first

w_{ρ}(z) = exp \{\frac{1}{2 π} ∫_{I} \frac{e^{it} + z}{e^{it} − z} log ρ(e^{it}) dt\} : z ∈ 𝔻 ,  (6)

for the outer H^{∞} function with modulus ρ ∈ L^{∞}(𝕋), ρ ≥ 0, and log ρ ∈ L^{1}(𝕋) with positive value at the origin. Defining then φ = w_{ϱ∨1} for the outer “quenching” function with constant modulus ϱ > 1 on I and 1 on J, we get that

g_{λ}(z) = f_{α}(z) := \frac{1}{2 i π} ∫_{I} (\frac{φ(ξ)}{φ(z)})^{α} f(ξ) \frac{dξ}{ξ − z}  for z ∈ 𝔻 ,  (7)
= \frac{1}{φ(z)^{α}} (P_{H^{2}} χ_{I} φ^{α} f)(z) ,  where α = −\frac{log(λ + 1)}{2 log ϱ} .

For f ∈ H_{|I}^{2} and the collection (f_{α}) of H^{2} functions defined by (7), it is already known from [3, Theorems 1.1, 1.11] that, as α → ∞:

(i) f_{α} converges to f in H^{2}; and hence (ii) f_{α}(z) tends to f(z) uniformly over compact subsets of 𝔻.

Point (i) is Patil’s result from [79] for p = 2 while point (ii) follows from the Goluzin–Krylov local convergence result, see also [78, 79]. However, these two points are improved by the analysis performed in [9] which is to the effect that, if f ∈ H_{|I}^{2}: (i)′ we have the following estimate of the H^{2} convergence rate: ∥f − f_{α}∥_{L^{2}(𝕋)} = O(exp(−α log ϱ)) → 0 as α → ∞; (ii)′ if f admits an absolutely continuous derivative on 𝕋, then f_{α}(z) tends to f(z) almost everywhere on 𝕋.

Point (i)′ follows from the estimate ∥f − g_{λ}∥_{L^{2}(𝕋)} = O((1 + λ)^{1/2}) → 0 as λ → −1, even without further assumptions on f. Point (ii)′ is obtained by further improving estimates of this H^{2} convergence rate under the additional smoothness hypothesis. This enables one to conclude that there is pointwise convergence of g_{λ} to f almost everywhere on 𝕋; we refer to [9] for further details. A further extension of point (i) comes from the fact that, for all f ∈ L^{2}(I), e(λ) = ∥f − g_{λ}∥_{L^{2}(I)} → 0 as λ → −1, see Section 3.1. Hence, even if f does not already belong to H_{|I}^{2}, (i)′′ f_{α} converges to f in L^{2}(I) as α goes to infinity.

Those considerations lead to the following characterization of traces on I of H^{2} functions, see [11]. Analogous results hold in general H^{p} spaces.

Construct for each f ∈ L^{2}(I) a family (g_{ρ}) of H^{2} functions such that g_{ρ |_{I}} converges to f in L^{2}(I) when ρ tends to some value ρ_{0} and g_{ρ} converges in H^{2} as ρ → ρ_{0} whenever f ∈ H_{|I}^{2}; then f ∈ H_{|I}^{2} if and only if ∥g_{ρ}∥_{L^{2}(J)} remains bounded.

If one takes for instance g_{ρ} = g_{λ} = f_{α} defined by (7), this is illustrated by the fact that, as λ → −1, either f ∉ H_{|I}^{2} and M(λ) → ∞, or f ∈ H_{|I}^{2} and M(λ) → ∥f∥_{L^{2}(J)} (point (i)), while ∥f − g_{λ}∥_{L^{2}(I)} → 0 in both cases (point (i)′′). The above principle can also be viewed as a consequence of [6, cor. 1] or [83, I, Theorem 1] which assert that a given function f ∈ L^{2}(I) is the trace of some H^{2} function of L^{2}(𝕋) norm less than M_{𝕋} if and only if, for all ϕ ∈ L^{2}(I),

|(f , ϕ)_{L^{2}(I)}| ≤ M_{𝕋} ∥P_{H^{2}} χ_{I} ϕ∥_{L^{2}(𝕋)} .

See also [66, Theorem 4.2] for an abstract version of this characterization result in the Hilbertian framework (3). Such issues are also examined in [6] for rather general classes of reproducing kernel Hilbert spaces of analytic functions, including Hardy–Sobolev spaces for example.

3.3. Hardy classes of circular domains and strips Let G be a bounded, finitely-connected domain in ℂ, whose boundary consists of the union of a finite number of disjoint Jordan loops. We know [45, Theorem 2, p. 237] that such a domain is conformally equivalent to a circular domain, that is, a domain consisting of the open unit disc from which a finite number of pairwise disjoint closed discs have been removed, and from now on we assume that G is itself a circular domain:

G = 𝔻 \ ⋃_{j=1}^{N}(a_{j} + r_{j}\overline{𝔻}),  (8)

with the obvious inequalities satisfied by the a_{j} and r_{j} for j = 1, …, N. We write D_{j} = a_{j} + r_{j}𝔻 for 1 ≤ j ≤ n. Let Γ denote the boundary of G, so that

Γ = Γ_{0} ∪ Γ_{1} ∪ … ∪ Γ_{N},

where Γ_{0} = ∂ 𝔻, and, for j = 1, …, N, we have Γ_{j} = a_{j} + r_{j}∂ 𝔻. We normalize the Lebesgue measure on Γ so that each circle Γ_{j} is given unit measure. For 1 ≤ j ≤ N, we write Ω_{j} = ℂ \ \{a_{j} + r_{j}\overline{𝔻}\}, so that G = 𝔻 ∩ ⋂_{j=1}^{N} Ω_{j}.

The Hardy spaces H^{p}(G) on a circular domain G were defined by Rudin [85] in terms of analytic functions f such that |f(z)|^{p} has a harmonic majorant on G, that is, a real harmonic function u(z) such that |f(z)|^{p} ≤ u(z) on G. Rudin then showed that H^{p}(G) has a natural decomposition into “elementary” Hardy spaces. That is, for 1 ≤ p < ∞ the Hardy space H^{p}(G) can equivalently be defined by

H^{p}(G) = H^{p}(𝔻) ⊕ ⨁_{j=1}^{N} H_{0}^{p}(Ω_{j}),  (9)

where for j ≥ 1 we define the Hardy space H_{0}^{p}(Ω_{j}) of functions analytic in Ω_{j}, with a zero limit at infinity in the obvious way. It is known (cf. [28]) that the sum in (9) is a topological direct sum, that is, that the projections onto the summands are bounded operators. It is also possible to define the Hardy spaces H^{p}(∂G) for 1 ≤ p < ∞ as the closure in L^{p}(∂G) of the set R_{G} of rational functions whose poles lie in the complement of \overline{G}. This approach, similar to one in [8], was taken in [28]. The spaces H^{p}(G) and H^{p}(∂G) are then isomorphic in a natural way, and so we identify the two spaces. In the case p = ∞, we can regard H^{∞}(G), the space of bounded analytic functions on G, as a direct sum of spaces H^{∞}(𝔻) ⊕ ⨁_{j=1}^{N} H_{0}^{∞}(Ω_{j}) (cf. Theorem 7.1 of [32]), and hence we may regard it as a closed subspace H^{∞}(∂G) of L^{∞}(∂G). We give the subspaces H^{p}(∂G) the norm inherited from the spaces L^{p}(∂G). The most completely analysed example of a circular domain is the annulus G = 𝔻 \ s𝔻 for some fixed s with 0 < s < 1. Here the Lebesgue measure on ∂G is normalized so that the circles 𝕋 and s𝕋 each have unit measure. It can be shown easily that the sequence of functions (e_{n})_{n∈ℤ}, defined by e_{n}(z) = z^{n}/\sqrt{1 + s^{2n}}, is an orthonormal basis of H^{2}(∂G).

Let us now work with the set-up (3), where H = L^{2}(∂G), H_{+} = H^{2}(∂G), H_{−} = H ⊖ H_{+}, K_{+} = L^{2}(𝕋) and K_{−} = L^{2}(s𝕋). The solution to the standard Bounded Extremal Problem 2.3 can now be expressed as follows.

Theorem 3.3. [90] Let f_{0} ∈ L^{2}(𝕋) \ P_{L^{2}(𝕋)}H^{2}(∂G), f_{1} ∈ L^{2}(s𝕋) and M > 0. Then there exists a function g_{0} ∈ H^{2}(∂G) such that ∥g_{0} − f_{1}∥_{L^{2}(s𝕋)} ≤ M and

∥f_{0} − g_{0}∥_{L^{2}(𝕋)} = inf{∥f_{0} − g∥_{L^{2}(𝕋)} : g ∈ H^{2}(∂G), ∥g − f_{1}∥_{L^{2}(s𝕋)} ≤ M}.

Indeed, if f_{0}(z) ∼ ∑_{n∈ℤ} a_{n}z^{n} for z ∈ 𝕋 and f_{1}(z) ∼ ∑_{n∈ℤ} b_{n}z^{n} for z ∈ s𝕋, then

g(z) = ∑_{n∈ℤ} \frac{a_{n} + αs^{2n}b_{n}}{1 + αs^{2n}} z^{n},

where α > 0 is the unique constant such that

∑_{n∈ℤ} \frac{|a_{n} − b_{n}|^{2}s^{2n}}{(1 + αs^{2n})^{2}} = M^{2}.

The key to the proof of the above theorem is that the Toeplitz operator P_{H_{+}}P_{K_{−}} on H_{+} is diagonalizable in a very simple way: the functions z^{n} are eigenvectors for n ∈ ℤ, and the corresponding eigenvalues are s^{2n}/(1 + s^{2n}). The analysis of such problems has applications in the theory of inverse Dirichlet– Neumann problems, such as those involved in the detection of faults in materials from electrical or thermal measurements. We refer to [12] for more details, see also Section 7. A similar analysis was performed in [90] for the Hardy space on the strip

X = {z ∈ ℂ : 0 < Re z < 1},

that is, taking H = L^{2}(∂X), H_{+} = H^{2}(∂X), the closure of the subspace consisting of rational functions with poles ℂ \ \overline{X}, K_{+} = L^{2}(iℝ) and K_{−} = L^{2}(1 + iℝ). An analogous extremal problem can be formulated and solved as a corollary to the results of [66] recalled in Section 2.2. In this case the Toeplitz operator T = P_{H_{+}}P_{K_{−}} on H_{+} can be written as T = L^{−1}M_{ϕ}L, where L is the bilateral Laplace transform, and M_{ϕ} is multiplication by the function ϕ(t) = e^{−t}/(e^{t} + e^{−t}) on L^{2}(ℝ). The spectrum of T is now the whole interval [0, 1], rather than a discrete set, and very explicit solutions to questions such as the corresponding Bounded Extremal Problem can be derived.

4. Applications to systems identification

Approximation in Hardy spaces (particularly H^{2} and H^{∞}) has been of great use recently in the identification of linear time-invariant systems. We recall that if T is a bounded operator on ℓ^{2} that commutes with the shift then, by means of the unitary z-transformation

Z : ℓ^{2} → H^{2}, Z((a_{n})) = ∑_{n=0}^{∞} a_{n}z^{n},

T can be regarded as a multiplication operator on the Hardy space H^{2}. More precisely, if T u = y, then Y (z) = G(z)U(z) for some G ∈ H^{∞} with ∥G∥_{∞} = ∥T ∥, where Y denotes Zy and U denotes Zu. Similarly, continuous time-invariant systems T (operators commuting with all right shifts) on L^{2}(0, ∞) correspond to multiplication operators on H^{2}(ℂ_{+}), the Hardy space of the right-hand half plane, by means of the Laplace transform,

L : L^{2}(0, ∞) → H^{2}(ℂ_{+}), Lf(s) = ∫_{0}^{∞} f(t)e^{−st} dt.

Again, if T u = y, then Y (s) = G(s)U(s), for some G ∈ H^{∞}(ℂ_{+}), with ∥G∥_{∞} = ∥T ∥, where Y denotes Ly and U denotes Lu. In the language of systems theory, u is called the input, y the output, and G the transfer function of the system. We refer to [41, 52] for introductions to this subject. See also [37, 50] for more on the links between operator theory and engineering and [44, 91] for the intriguing connexions between shift-invariant operators and transfer functions. The problem of identification is to obtain an approximation or model of an unknown system by means of a finite number of measurements. These can be input– output measurements (which we do not discuss here), or transfer function measurements, on the circle or imaginary axis as appropriate, which can be obtained by applying periodic inputs and measuring the corresponding outputs. The norms most commonly used for measuring the error of an approximating system are the H^{∞} and H^{2} norms. Worst-case identification assumes that the measured values are perturbed by disturbances (for example, nonlinearities or noise) which are small in some sense; it is to be contrasted with stochastic identification, which assumes some statistical model for the disturbances. We refer to the books [30, 78, 87] for more about worst-case identification, as well as to the survey articles [69, 71, 72]. Some early papers in the subject of worst-case frequency-domain identification are are [46, 49, 76, 77].

4.1. Solutions to uniform Bounded Extremal Problems We return to considerations about uniform meromorphic approximation issues of AAK type with constraint in modulus that generalize Problem 2.1 for p = ∞ and have been considered in [15]. Classes of uniform extremal problems are also considered in [51]. Define R_{N} ⊂ L^{∞}(𝕋) to be the set of rational functions with no poles on 𝕋 and at most N poles in 𝔻. For Ψ , M ∈ L^{∞}(J) and M > 0 a.e. on J, we define the subset B_{M,Ψ}^{N,∞} of H^{∞} + R_{N} by:

B_{M,Ψ}^{N,∞} := {g ∈ H^{∞} + R_{N} : |Ψ − g| ≤ M a.e. on J} .

Problem 4.1. For f ∈ L^{∞}(I), Ψ, M as above, find g_{N} ∈ B_{M,Ψ}^{N,∞} such that

∥f − g_{N}∥_{L^{∞}(I)} = min_{g∈B_{M,Ψ}^{N,∞}} ∥f − g∥_{L^{∞}(I)} := β_{N}(f ∨ ψ, M) .

Note that the particular case where N = 0 and M is constant coincides with Problem 2.1 for p = ∞. We now present a brief account of theoretical and computational results about Problem 4.1. Put C_{M,Ψ}^{N,∞} := {g_{|I} : g ∈ B_{M,Ψ}^{N,∞} } and let w_{β_{N} ∨M} ∈ H^{∞} be the outer function defined by (6).

Theorem 4.2. [15] If f ∨ Ψ ∈ H^{∞} + C(𝕋) , then

(i) there exists a solution g_{N} to Problem 4.1. If moreover f ∉ C_{M,Ψ}^{N,∞}, then: (ii) g_{N} is unique and |f−g_{N}| = β_{N} while |Ψ−g_{N}| = M a.e. on I and J respectively; (iii) we have g_{N} = v_{N} w_{β_{N} ∨M} for the solution v_{N} to the AAK problem:

∥(f ∨ Ψ)w_{β_{N} ∨M}^{−1} − v_{N}∥_{L^{∞}(𝕋)} = min_{v∈H^{∞}+R_{N}} ∥(f ∨ Ψ)w_{β_{N} ∨M}^{−1} − v∥_{L^{∞}(𝕋)} , (10)

whose value is equal to 1.

Again if N = 0 and M is constant, (i) and (ii) above correspond to Theorem 2.2, (ii). Note that the assumptions Ψ ∈ (H^{∞} + C(𝕋))_{|J} and M > 0 are sufficient to ensure that B_{M,Ψ}^{∞} ≠ ∅, see [13, Theorem 1]; hence B_{M,Ψ}^{N,∞} ≠ ∅. This need not be true in general. Observe also that β_{N} > 0, for f ∉ C_{M,Ψ}^{N,∞}; hence w_{β_{N} ∨M} is invertible in H^{∞}. Point (ii) is a direct consequence of (iii) together with properties of AAK problems on 𝕋 recalled in Section 2.1, that ensure existence and uniqueness of v_{N} in (10). Furthermore, under the above assumption, the Hankel operator Γ with symbol (f ∨ψ)w_{β_{N} ∨M}^{−1} (defined in (2)) is compact and the value of Problem (10) is equal to its (N +1)th singular value σ_{N+1}. Recall that, in this case, σ_{N+1} coincides with the positive square root of the (N + 1)st largest eigenvalue of the selfadjoint H^{2} operator Γ^{∗} Γ; call ν_{N+1} the associated eigenvector.

Theorem 4.2 (iii) efficiently reduces Problem 4.1 to AAK approximation and provides a way to solve it whenever f, Ψ, M satisfy the hypotheses.

1. Compute the unique positive value β_{N} such that σ_{N+1} = 1; this can be done by using a dichotomy procedure on the singular value computation.

2. Obtain the solution v_{N} to Problem (10):

v_{N} = (f ∨ ψ)w_{β_{N} ∨M}^{−1} − \frac{Γ ν_{N+1}}{ν_{N+1}} .

3. Build g_{N} accordingly to (iii).

This is almost a constructive procedure, except for the fact that AAK problems can be effectively solved only when the function to be approximated is rational, which is not the case of (f ∨ψ)w_{β_{N} ∨M}^{−1} in Problem (10). It raises a rather bad aspect of AAK problems which is the lack of uniform continuity of the associated best approximation operator [73]. In other spaces of functions, however, continuity does hold under some sufficient conditions [80], and this leads to a generic computational algorithm which is described in more detail in [15]. By performing a preliminary rational approximation step in the separable Hölder–Zygmund class, followed by the necessary AAK approximation step, and using Theorem 4.2, it is possible to solve Problem 4.1 in the generic sense that the algorithm is valid for an open dense subset of the possible data.

4.2. Robust identification from band-limited frequency data One well-established algorithm for frequency-domain identification of linear systems, which is an inverse problem for a convolution operator, uses a two-stage process [46, 49, 76, 77]. The measured data consists of a finite set of values, corresponding to corrupted measurements of an unknown transfer function h ∈ A(𝔻) at a finite set of points on the unit circle 𝕋. In the first step a continuous function f ∈ C(𝕋) is constructed, which interpolates the given data in an appropriate sense (for numerical purposes it is usual to work with a trigonometric polynomial). This is then followed by a Nehari approximation scheme, approximating f optimally by an H^{∞} function g. The validity of this approach follows from the easily-proven inequality

∥g − h∥_{L^{∞}(𝕋)} ≤ ∥g − f∥_{L^{∞}(𝕋)} + ∥f − h∥_{L^{∞}(𝕋)} ≤ 2∥f − h∥_{L^{∞}(𝕋)}.

In practical applications, h is the transfer function of a system that behaves linearly only in some frequency range, and approximate values of h may not be available at all parts of the unit circle. This problem is seen most clearly in the continuous-time case, when measurements are to be taken on the imaginary axis, and cannot be taken at arbitrarily high frequencies; by a conformal mapping, we can transform the mathematical analysis to the disc. In [16, 17] the techniques of band-limited approximation were applied to some data provided by the French National Space Agency (CNES). The algorithm is based on the following construction:

1. Given approximate values of an unknown function g ∈ A(𝔻) on a subinterval I of the circle, construct a smooth function f ∈ C(I) interpolating the data.

2. Given M > 0 solve the Bounded Extremal Problem 4.1 with N = 0, namely, find g_{0} ∈ H^{∞} such that ∥g_{0}∥_{L^{∞}(J)} ≤ M and

∥g_{0} − f∥_{L^{∞}(I)} = min{∥g − f∥_{L^{∞}(I)} : g ∈ H^{∞}, ∥g∥_{L^{∞}(J)} ≤ M}.

As this stands, the algorithm produces a model which is unsuitable for practical use (for instance, it is not normally in A(𝔻)), but various rational approximation steps can be used to circumvent this problem. Fuller technical details may be found in [16, 17].

5. Approximation with constraints on real or imaginary part

5.1. Passive systems We now give an account of another extremal problem with applications in Systems Theory, which was solved by McCarthy and Jacobson in [70] and also by Fisher and Micchelli in the unpublished report [38]. Let P denote the set of H^{2} functions with non-negative real part, i.e.,

P = {g ∈ H^{2} : Re g(z) ≥ 0 for all z ∈ 𝔻}.

The problem under consideration is as follows.

Problem 5.1. Given f ∈ H^{2}, find g_{0} ∈ P such that

∥f − g_{0}∥_{L^{2}(𝕋)} = inf{∥f − g∥_{L^{2}(𝕋)} : g ∈ P}.

This infimum is attained, and g_{0} is unique, because P is a closed convex subset of a Hilbert space. Let us write f and g_{0} in terms of their real and imaginary parts thus: f = p + iq, g_{0} = u + iv. Without loss of generality we may take q(0) = v(0) = 0. Let μ denote normalized Lebesgue measure on 𝕋.

Theorem 5.2. [70] Under the above conditions, the optimal approximant g_{0} = u+iv satisfies u = max(p + c, 0), where c = (1/2) ∫_{𝕋}(u − p) dμ is the unique point at which the decreasing function

x ↦ xμ{z : p(z) > −x} − ∫_{p≤−x} p dμ − 2x

on ℝ^{+} changes sign.

Given u we can compute v as the Hilbert transform of u. The authors of [70] give applications to the study of passive linear systems, which are those that can be modelled by transfer functions in P. The physical importance of such systems is explained in [34, 47]. Applications of the result above can be found to the model matching problem and systems identification.

5.2. Approximation with constrained imaginary part Consider now the following constrained approximation problem, where, as before, 𝕋 is written as the disjoint union of two sets I and J of positive measure.

Problem 5.3. Given f ∈ L^{2}(I), Ψ ∈ L^{2}(J) and M ≥ 0, find g_{0} ∈ H^{2} such that ∥ Im g_{0} − Ψ∥_{L^{2}(J)} ≤ M and

∥g_{0} − f∥_{L^{2}(I)} = min{∥g − f∥_{L^{2}(I)} : g ∈ H^{2}, ∥ Im g − Ψ∥_{L^{2}(J)} ≤ M}.

Boundary parameter identification for the 2D Laplace equation may be efficiently formulated as Problem 5.3 since the constraint can be used to impose the condition that the imaginary part of g_{0} should be close to some known function Ψ on J that may be available through boundary measurements, as in [12, 26], see Section 7. The above problem was analysed in [55], and its solution can be expressed implicitly in terms of the solution to the more standard Bounded Extremal Problem 3.1. Indeed, constraints on J in both problems are linked by ∥ Im g − Ψ∥_{L^{2}(J)} = ∥g−(Re g−iΨ)∥_{L^{2}(J)}; hence g_{0} is related to the solution to Problem 3.1 associated to the implicit gauge function Re g − iΨ on J. A direct solution can also be given in terms of the real-linear operator τ_{λ} : H^{2} → H^{2} defined (for λ > 0) by

τ_{λ}g = P_{H^{2}}(gχ_{I}) + iλP_{H^{2}}(Im gχ_{J}) .

It turns out that, for every λ > 0, τ_{λ} is a monotone operator (see, for example,

[24] for the properties of such operators) and hence is invertible .

Theorem 5.4. [55] Let M > 0 and suppose that f ∈ L^{2}(I) is not already the trace of an H^{2} function g with ∥ Im g − Ψ∥_{L^{2}(J)} ≤ M. Then the unique solution g_{0} to Problem 5.3 is given by

τ_{λ}g_{0} = P_{H^{2}}(f ∨ iλΨ),

for the unique λ > 0 such that ∥ Im g_{0} − Ψ∥_{L^{2}(J)} = M. Moreover, if f is the trace on I of an H^{2} function g with ∥ Im g − Ψ∥_{L^{2}(J)} ≤ M, then g_{0} = f and τ_{0}g_{0} = P (f ∨ 0).

Note that τ_{λ} actually involves both Toeplitz and Hankel operators, for we also have:

τ_{λ}g = g + (\frac{λ}{2} − 1) P_{H^{2}}(gχ_{J}) − \frac{λ}{2} P_{H^{2}}(\overline{g}χ_{J}) ,

where \overline{g} stands for the function z → \overline{g(z)}.

6. Applications to Signal Processing

Let Ω = [−b, b] and J = [−T, T] be subintervals of the real line. An important decomposition in the context of signal processing is the following:

L^{2}(ℝ) = P W (Ω) ⊕ P W (ℝ \ Ω) = L^{2}(J) ⊕ L^{2}(ℝ \ J),  (11)

where P W (Ω) is the Paley–Wiener space of band-limited signals, namely the subspace consisting of those functions f in L^{2}(ℝ) whose Fourier transform is supported on Ω, that is,

f(t) = \frac{1}{2π} ∫_{Ω} f̂(ω)e^{iωt} dt,

where f̂ ∈ L^{2}(Ω). See [78], for example. More generally, let G be any locally compact Abelian group with dual group Γ, and let F : L^{2}(G) → L^{2}(Γ) be the Fourier–Plancherel transform, which can be defined as a unitary operator [86]. We may consider the decomposition of the form (3), where H = L^{2}(G), H_{+} = F^{−1}L^{2}(A), H_{−} = F^{−1}L^{2}(Γ \ A), K_{+} = L^{2}(B), and K_{−} = L^{2}(G \ B), for fixed Borel sets A ⊆ Γ and B ⊆ G. The spectral theory of the corresponding Toeplitz operators is simpler in this case.

Theorem 6.1. [90] Suppose that A and B have finite Haar measure; then the Toeplitz operator S := P_{H_{+}}P_{K_{+}} defined on H_{+} is Hilbert–Schmidt, and hence T := P_{H_{+}}P_{K_{−}} has an orthonormal basis of eigenvectors.

In the case of the Paley–Wiener decomposition (11), the eigenvectors of T are known as Prolate Spheroidal Wave Functions, and are described in [74, 89]. They form an orthonormal basis (ψ_{n}) of P W (Ω), and (ψ_{n}χ_{J}/\sqrt{λ_{n}}) forms an orthonormal basis of L^{2}(J), where (λ_{n}) is the sequence of eigenvalues of S. This enables us to provide an explicit solution to various problems in signal analysis, linked with approximating time-limited signals by the restriction of bandlimited signals (see [88] for an interesting discussion of such problems). The following result, due originally to Slepian and Pollak [89] and derived by Smith [90], along with additional results on constrained approximation, as a corollary of the results of [66], is typical.

Theorem 6.2. Suppose that f ∈ L^{2}(J) \ P_{L^{2}(J)}P W (Ω) and M > 0. Then there exists g_{0} ∈ P W (Ω) such that ∥g_{0}∥_{L^{2}(ℝ\J)} ≤ M and

∥g_{0} − f∥_{L^{2}(J)} = inf{∥g − f∥_{L^{2}(J)} : g ∈ P W (Ω), ∥g∥_{L^{2}(ℝ\J)} ≤ M}.

Moreover, writing f = ∑_{n=0}^{∞} a_{n}ψ_{n}χ_{J} for some coefficients (a_{n}), then we have

g_{0} = ∑_{n=0}^{∞} \frac{λ_{n}a_{n}}{μ(1 − λ_{n}) + λ_{n}} ψ_{n},

where μ > 0 is the unique constant such that

∑_{n=0}^{∞} \frac{(1 − λ_{n})λ_{n}^{2}|a_{n}|^{2}}{(μ(1 − λ_{n}) + λ_{n})^{2}} = M^{2}.

A good general reference for Toeplitz and Hankel operators on Paley–Wiener spaces is [82].

7. Applications to 2D inverse problems for Laplace operator

For various partial differential equations, inverse problems that consist in recovering missing data on some part J of the boundary ∂D of a domain D from data on the complementary part I = ∂D \ J arise in many applications of nondestructive control (tomography and medical imaging, fault detection and control, underground prospecting, etc.), see [54]. Particular two-dimensional instances of such issues, namely the ones for D ⊂ ℂ that involve the Laplace operator and harmonic functions (see e.g. [4, 7]), can be efficiently rephrased in terms of constrained extremal problems in spaces of analytic or meromorphic functions as in [12, 26]. In order to illustrate this approach, we consider here the following inverse problem of parameter identification in 𝔻, see [26]. Note that, by making a conformal transformation, it can be handled in general simply connected Hölder-smooth domains D. Such an issue arises for example in corrosion detection by electrical impedance tomography. In the simple linear case, the parameter characterizing the damage caused by corrosion is a Robin exchange coefficient, the direct problem to be solved being a Laplace equation.

Problem 7.1. Given a prescribed (current) flux ϕ ∈ L^{2}(I) , ϕ ≢ 0 together with measurements (of the voltage potential) u_{m} ∈ W^{1,2}(I), find a function φ on J such that the solution u of

\{ \begin{matrix}Δ u & = & 0 & in & 𝔻, \\ \frac{∂u}{∂n} & = & ϕ & on & I, \\ \frac{∂u}{∂n} + φu & = & 0 & on & J, \end{matrix}  (12)

also satisfies u_{|_{I}} = u_{m}.

We stick here to the classical notation where n denotes the outer unit normal vector to 𝕋. Smoothness and identifiability (uniqueness) results from [25, 26] ensure that if ϕ ∈ L^{2}(I) and φ_{1}, φ_{2} belong to some class of continuous functions on J, then the associated solutions u_{1}, u_{2} of (12) belong to the Sobolev space W^{1,2}(𝕋) of functions that admit an L^{2}(𝕋) derivative; moreover, it holds that if u_{1|_{I}} = u_{2|_{I}}, then φ_{1} = φ_{2} on J. Solving the Inverse Problem 7.1 with the use of tools from Section 3 first requires us to go from harmonic to analytic functions. From the knowledge of the flux ϕ ∈ L^{2}(I) and of the voltage potential u_{m} ∈ W^{1,2}(I) in system 7.1, we can build the trace on I of a function g analytic in 𝔻. Indeed,

f = u_{m} + i ∫ ϕdθ  on I,  (13)

is the trace on I of a (unique) function g analytic in 𝔻: f = g_{|I}. This holds because any harmonic conjugate function v to u in 𝔻 (such that f = u + iv is an analytic function in 𝔻) satisfies the Cauchy–Riemann equations:

\frac{∂v}{∂θ} = \frac{∂u}{∂n} on 𝕋 .

That g actually belongs to the Hardy–Sobolev H^{2} ∩ W^{1,2}(𝕋) follows from smoothness preserving properties of the Hilbert transform (harmonic conjugation operator), for instance the M. Riesz theorem, see [2, 21, 43]. Our aim is then to recover g on J from the knowledge of its trace f on I. This would solve Problem 7.1, since

u = Re g in 𝔻,

and then

φ = − \frac{1}{Re g} \frac{∂ Im g}{∂θ}  on J ,  (14)

is the expected solution. Note that the above equality should be properly understood (non-tangential limits on the right hand side). However, we want the recovery procedure to be convergent in order to carry some stability and robustness properties. Our concern here is that, in practice, one may not know exactly u_{m} nor f on I: values of u_{m} might only be available through experimental devices that necessarily induce perturbations while the knowledge of f also requires that of a primitive of ϕ, which may be computed numerically. Hence, f does not coincide in general with the trace on I of an H^{2} function. This is the reason why the following approach is taken, with a resolution algorithm which solves a Bounded Extremal Problem rather than an extrapolation problem [26].

1. From the data u_{m} and ϕ, build f on I through (13).

2. Solve Problem 3.1 for f, Ψ = 0, and a suitable value of the constraint M (it can be computed by cutting I into two disjoint parts I_{1}, I_{2} and iteratively solving a bounded extremal problem on I_{1}, in order to minimize a criterion on I_{2}).This gives a function g_{2}.

3. Finally, applying (14) to g_{2} gives the desired coefficient φ. In fact, two additional steps around Step 2 above are needed in order for the above algorithm to converge, which is in accordance with a global stability result concerning Problem 7.1, [26]. Indeed, the procedure has to be applied at a higher order: Step 2 must be applied to the tangential derivative of f on I and be followed by the computation of an appropriate primitive g_{2} of the solution to Problem 3.1. Note that this is a way to solve Bounded Extremal Problems in Hardy–Sobolev spaces [10], and that continuity properties can be deduced from boundary estimates in these classes, see [20]. Various other 2D inverse problems can be handled by analytic, meromorphic, or rational approximation techniques, such as crack detection from boundary data as in [12], where it shown how approximants provide information about the singularities of the solution. Also, Hardy spaces of circular domains, see Section 3.3, may be of use to model the presence of inclusions or defaults in some material. Concerning for instance the recovery of a damaged and unknown part of the boundary, extremal problems with constraint on the imaginary part, see Section 5.2, may furnish a suitable framework, since this imaginary part (flux) may be data of the problem on the whole boundary.

8. Interpolation between H^{2}(∂G) and H^{∞}(∂G) with applications

8.1. Interpolation spaces In this section we observe that the H^{p}(∂G) spaces (2 < p < ∞), defined in Section 3.3, are interpolation spaces between H^{2}(∂G) and H^{∞}(∂G) in a sense to be made precise. This result is a version of the celebrated theorem of Jones [57] that established the same result for the disc. Other important interpolation results in this area are due to Pisier [81] and Bourgain [23]. Proving versions of the Jones result valid for H^{p}(G) (and there seems to be no way to deduce them directly), can be achieved by making constructions similar to one found in the work of Kisliakov and Xu [62, 63].

Theorem 8.1. Suppose that G is the annulus {z ∈ ℂ : s < |z| < 1}, for fixed s > 0. Let f ∈ H^{2}(∂G). Suppose that f = f_{0} + f_{1} where f_{0} ∈ L^{2}(∂G) and f_{1} ∈ L^{∞}(∂G). Then, for any μ > 0, there exist g_{0} ∈ H^{2}(∂G) and g_{1} ∈ H^{∞}(∂G) satisfying f = g_{0} + g_{1} with

∥g_{0}∥_{H^{2}(∂G)} ≤ (2 + 4μ)∥f_{0}∥_{L^{2}(∂G)} and ∥g_{1}∥_{H^{∞}(∂G)} ≤ (1 + 1/μ)∥f_{1}∥_{L^{∞}(∂G)}.

We may also generalize Theorem 8.1 to arbitrary circular domains although the precise constants involved are no longer so simple.

Theorem 8.2. Suppose that G is a circular domain defined by (8). For each μ > 0, there exists a positive constant C(G, μ) with the following properties. Let f ∈ H^{2}(∂G), and suppose that f = f_{0} + f_{1} where f_{0} ∈ L^{2}(∂G) and f_{1} ∈ L^{∞}(∂G). Then there exists g_{0} ∈ H^{2}(∂G) and g_{1} ∈ H^{∞}(∂G) satisfying f = g_{0} + g_{1} with

∥g_{0}∥_{H^{2}(∂G)} ≤ (2 + C(G, μ))∥f_{0}∥_{L^{2}(∂G)}

and

∥g_{1}∥_{H^{∞}(∂G)} ≤ (1 + 1/μ)∥f_{1}∥_{L^{∞}(∂G)}.

We can reformulate the above results in the language of interpolation spaces. Let A_{0}, A_{1} be a couple of interpolation (Banach) spaces. For x ∈ A_{0} + A_{1} and t > 0, let

K_{t}(x, A_{0}, A_{1}) = inf{∥x_{0}∥_{A_{0}} + t∥x_{1}∥_{A_{1}} : x = x_{0} + x_{1}, x_{0} ∈ A_{0}, x_{1} ∈ A_{1}}.

The real interpolation space (A_{0}, A_{1})_{θ,p} is defined to be the space of all x ∈ A_{0}+A_{1} such that ∥x∥_{θ,p} < ∞ where

∥x∥_{θ,p} = (∫_{0}^{∞}(t^{−θ}K_{t}(x, A_{0}, A_{1}))^{p} \frac{dt}{t})^{1/p}.

This notion is motivated by the well-known result L^{p} = (L^{2}, L^{∞})_{θ,p} with 2 < p < ∞ and \frac{1}{p} = \frac{1−θ}{2} , [22, Section 5.2]. Peter Jones [57] gave the following interpolation results for the unit circle:

H^{p} = (H^{2}, H^{∞})_{θ,p},  where 2 < p < ∞ and  \frac{1}{p} = \frac{1 − θ}{2} ,

which we now extend to H^{p}(∂G).

Proposition 8.3. Let 2 < p < ∞ and f ∈ H^{p}(∂G). Then there exists a positive constant C_{G} such that

C_{G}K_{t}(f, H^{2}(∂G), H^{∞}(∂G)) ≤ K_{t}(f, L^{2}(∂G), L^{∞}(∂G))
≤ K_{t}(f, H^{2}(∂G), H^{∞}(∂G)),

and hence

H^{p}(∂G) = (H^{2}(∂G), H^{∞}(∂G))_{θ,p},  where  \frac{1}{p} = \frac{1 − θ}{2} .

8.2. Simultaneous H^{2} and H^{∞} approximation The first application of H^{2}(G)–H^{∞}(G) interpolation is in providing a very explicit version of the Nehari approximation theorem in order to obtain an analytic approximant to a given function which is to be nearly optimal in both L^{∞} and L^{2} norms. In the case of the disc, such approximants have significant implications in Systems and Control Theory, and were first provided by Kaftal, Larson and Weiss [58] in an operator-algebraic context. Further extensions were given in [39, 40, 42] by Frazho and his co-workers. The results have been expressed in terms of central interpolants and the more abstract context of the Commutant Lifting theorem. The following is a key result along these lines.

Theorem 8.4. [58] Given f ∈ L^{∞}(𝕋) and a number δ > 1, one can find a function h in H^{∞} such that simultaneously

∥f − h∥_{L^{∞}(𝕋)} ≤ δ dist_{∞}(f, H^{∞})

and

∥f − h∥_{L^{2}(𝕋)} ≤ \frac{δ}{\sqrt{δ^{2} − 1}} dist_{2}(f, H^{2}),

where dist_{∞}(f, H^{∞}) is the distance of f to H^{∞}, and similarly dist_{2}(f, H^{2}) is the distance of f to H^{2}.

The extensions we discuss here are for Hardy spaces on circular domains.

Theorem 8.5. Suppose that G is a circular domain defined by (8). For each μ > 0, there exists a positive constant C(G, μ) with the following properties. Let f ∈ L^{∞}(∂G). Then there exists g ∈ H^{∞}(∂G) such that

∥f − g∥_{L^{∞}(∂G)} ≤ (2 + 1/μ) dist_{∞}(f, H^{∞}(∂G))

and

∥f − g∥_{L^{2}(∂G)} ≤ (3 + C(G, μ)) dist_{2}(f, H^{2}(∂G)).

Remark that in the particular case where G is an annulus, we can take C(G, μ) = 4μ.

8.3. “Correction” results

The second application of Theorem 8.2 is in providing so-called “correction” theorems, analogous to those of of Kisliakov [60, 61], whereby an H^{p} function for 2 < p < ∞ is approximated by an H^{∞} function with small H^{2} error.

Theorem 8.6. Suppose that G is a circular domain defined by (8). Then there exists a constant K_{G} with the following properties. Let p > 2 and f ∈ H^{p}(∂G). For any ε > 0 there exist g_{0} ∈ H^{2}(∂G) and g_{1} ∈ H^{∞}(∂G) such that f = g_{0} + g_{1},

∥g_{0}∥_{H^{2}(∂G)} ≤ ε,  and  ∥g_{1}∥_{H^{∞}(∂G)} ≤ 3(\frac{2 + K_{G}}{ε} )^{2/(p−2)}.

It is also possible to give a direct proof of Theorem 8.6 which does not rely on Theorem 8.2. This is of particular interest in the case of the unit disc and we obtain thereby an improvement in the constants involved (cf. Proposition 2.2 of [27] which applies only for p > 4).

Proposition 8.7. [29, Prop. 3.1] Let p > 2 and f ∈ H^{p}. For any ε > 0 there exist g_{0} ∈ H^{2} and g_{1} ∈ H^{∞} such that f = g_{0} + g_{1},

∥g_{0}∥_{H^{2}} ≤ ε,  and  ∥g_{1}∥_{H^{∞}} ≤ (\frac{3}{ε})^{2/(p−2)} .

The following corollary of Theorem 8.6 can also be proved directly and in a stronger form (with (1/ε)^{δ} replaced by log(1/ε) in (15)) using a more general result of Kisliakov [60, 61]: these rely on weak-type (1 − 1) estimates for operators given by Calderón–Zygmund kernels.

Corollary 8.8. [28, Cor. 2.1] Let ε > 0 be sufficiently small, let δ > 0, and let f ∈ L^{∞}(∂G). Then there exist g^{+} ∈ H^{∞}(G) and g^{−} ∈ H^{∞}(ℂ \ \overline{G}) such that

∥f − (g^{+} + g^{−})∥_{L^{2}(∂G)} < ε∥f∥_{L^{∞}(∂G)}

and

∥g^{+}∥_{H^{∞}(∂G)} + ∥g^{−}∥_{H^{∞}(ℂ\\overline{G})} < C(1/ε)^{δ}∥f∥_{L^{∞}(∂G)} ,  (15)

where C is a constant that depends only on G and δ.

8.4. Factorization results Let H be a separable, infinite dimensional, complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Suppose that T ∈ L(H) is such that its spectrum is included in G^{−} (the closure of G). Denote by R_{G} the set of rational functions whose poles lie in the complement of G^{−}. Suppose also that

∥f(T)∥ ≤ sup{|f(λ)| : λ ∈ G}, f ∈ R_{G}.

By virtue of [35], we may assume without loss of generality with respect to properties we consider that T admits a G-rational normal boundary dilation, that is, there exists a Hilbert space K ⊃ H, a normal operator N ∈ L(K) such that σ(N) ⊂ ∂G and for any r ∈ R_{G} and any x ∈ H, r(T)x = P_{H}r(N)x. In the particular case where N is absolutely continuous (i.e., its spectral measure is absolutely continuous with respect to Lebesgue measure), we say that T ∈ DN_{a}(G). Let β_{0} denote the position function (i.e., β_{0}(z) = z) in any of the spaces we consider. Any absolutely continuous normal operator N ∈ L(K) is (unitarily equivalent to) a part of the operator M_{β_{0}} of multiplication by β_{0} on some space L^{2}(Γ, D) of D-valued functions on the Borel set Γ (D being a Hilbert space). For x, y ∈ K, one defines the function x ·^{T} y by x ·^{T} y(ξ) = (x(ξ), y(ξ))_{D}. Thus for T ∈ DN_{a}(G) we can define a sesquilinear map (x, y) ∈ H × H → x ·^{T} y ∈ L^{1}(Γ) which does not depend on any choice of the normal boundary dilation [31, p. 347]. We denote by P_{∗}(Γ) the predual of H^{∞}(Γ): P_{∗}(Γ) = L^{1}(Γ)/^{⊥}H^{∞}(Γ) and, for f ∈ L^{1}(Γ), by [f] the image of f in P_{∗}(Γ) via the quotient map. Using Corollary 8.8 one can prove the following result which generalizes [27, Theorem 3.2].

Theorem 8.9. [28, Thm 3.1] Let T ∈ DN_{a}(G) and let (x_{n})_{n≥1} be a sequence of elements of H. The following assertions are equivalent:

(i) lim_{n→∞} ∥x_{n} ·^{T} w∥_{L^{1}(Γ)} = 0,  w ∈ H
(ii) lim_{n→∞}(∥[x_{n} ·^{T} w]∥_{P_{∗}(Γ)} + ∥[w ·^{T} x_{n}]∥_{P_{∗}(Γ)}) = 0, w ∈ H.

Denote by Φ_{T} the H^{∞}(Γ) functional calculus for T ∈ DN_{a}(G) defined by Φ_{T} (h) = h(T), where

h(T)x = P_{H}h(N)x, h ∈ H^{∞}(Γ), x ∈ H.

Though N is not necessarily unique, this functional calculus is uniquely defined as shown by Theorem 2.3 in [31].

Definition 8.10. For a circular domain G, the class 𝔸^{G} is the subclass of DN_{a}(G) consisting of those T ∈ L(H) for which Φ_{T} is isometric. The class 𝔸_{ℵ_{0}}^{G} consists of those T ∈ 𝔸^{G} such that, for any infinite array (f_{i,j})_{i,j≥1} of functions in L^{1}(Γ), there exist some sequences (x_{i})_{i≥1} and (y_{j})_{j≥1} in H such that:

[f_{i,j}] = [x_{i} ·^{T} y_{j}], i, j ≥ 1.

Our final result generalizes Corollary 4.4 in [27] and is an immediate consequence of Theorem 8.9.

22 I. Chalendar, J. Leblond, and J. R. Partington

Corollary 8.11. [28, Cor. 3.1] Let T ∈ L(H) be in the class 𝔸_{ℵ_{0}}^{G}. Then, for any infinite array of functions in L^{1}(Γ), there exist bounded sequences (x_{i})_{i≥1} and (y_{j})_{j≥1} in H such that:

f_{i,j} = x_{i} ·^{T} y_{j}, i, j ≥ 1.

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Institut Girard Desargues, UFR de Mathématiques, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France

E-mail address: chalenda@desargues.univ-lyon1.fr

INRIA, BP 93, 2004 route des Lucioles, 06902 Sophia–Antipolis Cedex, France

E-mail address: leblond@sophia.inria.fr

School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.

E-mail address: J.R.Partington@leeds.ac.uk