Georgian Mathematical Journal Volume 8 (2001), Number 2, 275–281

PROPER MOVING AVERAGE REPRESENTATIONS AND OUTER FUNCTIONS IN TWO VARIABLES

L. GAWARECKI, V. MANDREKAR, AND P. RICHARD

Abstract. In this work, we consider the problem of moving average representations for random fields. As in the Kolmogorov–Wiener case, such representations lead to interesting questions in harmonic analysis in the polydisc. In particular, we study outer functions with respect to half-space, semigroup and quarterplane and their interrelations.

2000 Mathematics Subject Classification: 42B30, 60G10, 60G60. Key words and phrases: Outer functions, proper moving average representations, stationary random fields.

1. Introduction

Professor N. Vakhania has been interested in the area of Prediction Theory and Analysis for infinite dimensional stationary processes, [1]. In this work we study the moving average (MA) representations for weakly stationary random fields under half–spaces, semigroups and quarter–planes. We relate the properness of the MA representation to the analytic properties of the factor of the spectral density. We interpret the results of Helson and Lowdenslager, [3], in this context, and relate them to the H-outer property. Using a result in [3], one can show the equivalence of the outer property of the factor, and the properness of the semigroup induced MA representation. We use this result to relate λouter functions (in one variable, with the other variable acting as a parameter), and half-space MA representations.

Finally, we take up the quarter-plane MA representation and relate its properness to H-outer property of the factor. This can be exploited to obtain a probabilistic result of Soltani, [9]. Connection between outer functions in the sense of [3] and outer functions in H^{2} (T^{2}) is used to obtain the analytic results of Izuchi and Matsugu, [5], with very simple proofs. Throughout the paper, we use the standard terminology of the book [2].

2. Half-Space MA Representations

Let {Ω, F, P } be a probability space, and L^{2}(Ω, F, P) be the complex Hilbert space of (equivalence classes of) P -square integrable complex-valued functions. A family {X_{t}, t ∈ Z^{d}} ⊂ L^{2}(Ω, F, P) is called a weakly stationary random field

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

if E (X_{t}) = c (from now on c = 0), and

E (X_{t}X_{t′}) = R(t − t′).

In the case of d = 1, {X_{n}, n ∈ Z} is called a weakly stationary process. With {X_{n}}, one associates a spectral measure F through the Bochner Theorem,

R(n) = \frac{1}{2π} ∫_{−π}^{π} e^{inλ} dF (λ).

A stationary process has a MA representation if

X_{n} = ∑_{k=0}^{∞} a_{k}ξ_{n−k},  (2.1)

where {ξ_{k}, k ∈ Z} are orthonormal elements of L^{2}(Ω, F, P). It is well known by a result of Kolmogorov and Wiener, that X_{n} has a MA representation if and only if F ≪ Leb, and the density is given by f (e^{iλ}) = |φ (e^{iλ})|^{2}, where

φ (e^{iλ}) = ∑_{k=0}^{∞} ā_{k}e^{ikλ}.

From the point of view of prediction, one needs to know when the MA representation (2.1) is proper, i.e., when

H(X : n) = H(ξ : n), for each n.  (2.2)

Here, H(X : n) = \overline{sp} \{X_{k}, k ≤ n\}.

It is obvious that the above condition holds true if and only if φ (e^{iλ}) is H-

outer, i.e., \overline{sp} \{e^{inλ}φ (e^{iλ}) : n ≥ 0\} = H^{2} (T), where H^{2}(T) = \overline{sp} \{e^{inλ}, n ≥ 0\}, and the closure refers to the space L^{2}([−π, π], Leb). An analytic condition for φ to be H-outer is given by

log |∫_{−π}^{π} φ (e^{iλ}) dσ| = ∫_{−π}^{π} log |φ (e^{iλ})| dσ,

where σ is the normalized Lebesgue measure on [−π, π].

In case G = Z^{2}, Helson and Lowdenslager ([3]) considered the analogue of this problem by putting an ordering on Z^{2} induced by a semigroup S so that S ∪ −S = Z^{2}, and S ∩ −S = {(0, 0)}. A particular example of S is

S = {(j, k) : j ∈ Z_{+} for k = 0 and j ∈ Z for k ≥ 1}.  (2.3)

They showed that a stationary random field has a MA representation, i.e.,

X_{m,n} = ∑_{(j,k)∈S} a_{j,k}ξ_{m−j,n−k},

with {ξ_{j,k}}_{(j,k)∈Z^{2}}, orthonormal elements in L^{2}(Ω, F, P) if and only if the spectral measure of {X_{m,n}} on T^{2} is absolutely continuous with respect to σ_{2}, the normalized Lebesgue measure on T^{2}, and f (e^{iλ}, e^{iμ}) = |φ (e^{iλ}, e^{iμ})|^{2}. Here

φ (e^{iλ}, e^{iμ}) = ∑_{(j,k)∈S} ā_{j,k}e^{ijλ+ikμ} is of analytic type (in the sense of S).

Using the ordering on Z^{2} induced by S, i.e., (j, k) < (j′, k′) if (j′ −j, k′ −k) ∈ S, we define

H (X : (m, n)) = \overline{sp} \{X_{j,k} : (j, k) < (m, n) ∈ Z^{2}\} .

We say that a MA S-representation is proper if

H (X : (m, n)) = H (ξ : (m, n)) , for all (m, n) ∈ Z^{2}.

From the result in [3] we get that a MA representation is proper if and only if

log |∫_{T^{2}} φ (e^{iλ}, e^{iμ}) dσ_{2}| = ∫_{T^{2}} log |φ (e^{iλ}, e^{iμ})| dσ_{2},  (2.4)

where φ is the function of S-analytic type associated with the MA S-represen-tation. A function φ of S-analytic type is called outer if it satisfies equation (2.4).

Given a set A ⊂ Z^{2}, and g ∈L^{2} (T^{2}, σ_{2}), denote [A] :=\overline{sp} \{e^{ijλ+ikμ} : (j, k)∈A\},

and [g]_{A} := \overline{sp} \{e^{ijλ+ikμ}g : (j, k) ∈ A\}, closed linear subspaces of L^{2} (T^{2}, σ_{2}). The following result is a consequence of Theorem 6 in [3].

Theorem 2.1. Let φ be of S-analytic type. Then the following statements are equivalent:

(i) φ is outer, (ii) [φ]_{S} = [S].

Let us denote by A_{λ} := Z × Z_{+} the half-space in Z^{2}, and H_{λ}^{2} := [A_{λ}]. The condition (ii) of Theorem 2.1 implies that an outer function φ of S-analytic type satisfies

H_{λ}^{2} = [A_{λ}] = [[S]]_{A_{λ}} = [[φ]_{S}]_{A_{λ}} = [φ]_{A_{λ}},

giving [φ]_{A_{λ}} = H_{λ}^{2} .

Following [5], we define a function g ∈ L^{2}(T^{2}, σ_{2}) as λ-outer if the cut function g_{λ} (e^{iμ}) = g (e^{iλ}, e^{iμ}) is outer in the variable e^{iμ}, σ-a.e. (in e^{iλ}).

The following lemma is a consequence of a result in [4].

Lemma 2.2. If g ∈ H_{λ}^{2}, then g is λ-outer if and only if [g]_{A_{λ}} = H_{λ}^{2}.

Proof. By Theorem 2 in [4], [g]_{A_{λ}} = qH_{λ}^{2}, where q is a unimodular function, such that the cut function q_{λ} of q is inner in e^{iμ}. Since g = qh, and the cut function of g is outer, we get that q is constant in e^{iμ}, σ-a.e. Thus qH_{λ}^{2} = H_{λ}^{2}, giving the necessity.

To get the sufficiency, we assume without lost of generality that g (e^{iλ}, e^{iμ}) is in H^{2} (T_{μ}) for all λ, and as in [2], for each λ, we define

G_{r} (e^{iλ}, e^{iμ}) = exp (∫_{T} \frac{e^{iλ′} + re^{iμ}}{e^{iλ′} − re^{iμ}} log |g (e^{iλ}, e^{iμ})| dσ(λ′)) .

Then for all e^{iλ}, lim_{r→1^{−}} G_{r} (e^{iλ}, e^{iμ}) = G (e^{iλ}, e^{iμ}) is an outer factor (in e^{iμ}) of g as a function of e^{iμ}. Since

∫_{T^{2}} |G|^{2}dσ_{2} = ∫_{T^{2}} |g|^{2}dσ_{2} < ∞,

the function G ∈ L^{2} (T^{2}, σ_{2}), we obtain that G is λ-outer. Then q = g/G is inner in e^{iμ} for all λ. Hence, [g]_{A_{λ}} = q[G]_{A_{λ}}. From the proof of the necessity, [G]_{A_{λ}} = H_{λ}^{2}. Thus H_{λ}^{2} = [g]_{A_{λ}} = qH_{λ}^{2} and consequently, for each λ, q is constant in μ. Thus g = qG is λ-outer.

As a consequence, we obtain the following theorem.

Theorem 2.3. Let g be of S-analytic type. If g is outer, then g is λ-outer.

Given a function g of S-analytic type, we have

g (e^{iλ}, e^{iμ}) = ∑_{j=0}^{∞} a_{j,0}e^{ijλ} + ∑_{k≥1} ∑_{j∈Z} a_{j,k}e^{ijλ+ikμ}.

Let

h(e^{iλ}) = ∑_{j=0}^{∞} a_{j,0}e^{ijλ},  (2.5)

and observe that

∫_{T} g (e^{iλ}, e^{iμ}) dσ(μ) = h (e^{iλ}) .

Thus if h is outer, then

log |∫_{T^{2}} g (e^{iλ}, e^{iμ}) dσ_{2}| = log |∫_{T} ∫_{T} g (e^{iλ}, e^{iμ}) dσ(μ)dσ(λ)|

= ∫_{T} log |∫_{T} g (e^{iλ}, e^{iμ})| dσ(λ).

If further g is λ-outer, then log |∫_{T} g (e^{iλ}, e^{iμ}) dσ(μ)| = log ∫_{T} |g (e^{iλ}, e^{iμ})| dσ(μ). Hence we get

Lemma 2.4. Let g be of S-analytic type, and h be as in (2.5). If h is outer and g is λ-outer, then g is outer.

3. Quarterplane MA Representations

In [6], a quarterplane MA (QMA) representation was studied for G = Z^{2}. Here, one does not have an ordering on Z^{2} (except lexicographic). We say that {X_{m,n}, (m, n) ∈ Z^{2}} has a QMA representation if

X_{m,n} = ∑_{k≥0} ∑_{j≥0} a_{j,k}ξ_{m−j,n−k},  (3.1)

where {ξ_{j,k}, (j, k) ∈ Z^{2}} is an orthonormal family. We say that a function φ ∈ H^{2} (T^{2}) if

φ (e^{iλ}, e^{iμ}) = ∑_{j≥0} ∑_{k≥0} b_{j,k}e^{ijλ+ikμ}.  (3.2)

We note that φ ∈ H^{2} (T^{2}) implies that φ is of S-analytic type. It is easy to verify that {X_{m,n}} has the QMA representation (3.1) if and only if its spectral measure F ≪ σ_{2}, and the density f (e^{iλ}, e^{iμ}) = |φ (e^{iλ}, e^{iμ})|^{2}, where φ ∈ H^{2} (T^{2}) and

φ (e^{iλ}, e^{iμ}) = ∑_{j,k≥0} ā_{j,k}e^{ijλ+ikμ}.

A QMA representation is proper if and only if

H (X : (m, n)) = \overline{sp} \{X_{j,k} : j ≤ m, k ≤ n\} = H (ξ : (m, n)) .

For any second order random field {y_{m,n} : (m, n) ∈ Z^{2}}, we denote

L^{1}(y : m) = \overline{sp} \{y_{j,k} : j ≤ m, k ∈ Z\}
and L^{2}(y : n) = \overline{sp} \{y_{j,k} : j ∈ Z, k ≤ n\} .

Also, let p^{i}(y : m) be the projection onto L^{i}(y : m), i = 1, 2. We shall drop the dependence on y when it is clear from the context. We observe that for {ξ_{j,k}, (j, k) ∈ Z^{2}},

p^{1}(m)p^{2}(n) = p(m, n),  (3.3)

where p(m, n) is the projection on H (ξ : (m, n)). Thus we obtain that if the QMA representation (3.1) is proper, i.e., H (X : (m, n)) = H (ξ : (m, n)), then for the process {X_{m,n} : (m, n) ∈ Z^{2}}, equality (3.3) holds true. This condition was introduced in [6]. Denote, as in [6], L^{i}(X : −∞) = ⋂_{m} L^{i}(X : m), i = 1, 2. It was further proved in [6] that under condition (3.3), and the condition:

\overline{sp} \{L^{1}(X : −∞) ∪ L^{2}(X : −∞)\} = \{0\},  (3.4)

a weakly stationary random field has a proper QMA representation. We observe

Theorem 3.1. The QMA representation (3.1) for {X_{m,n}} is proper if and only if the following three conditions are satisfied:

(i) the spectral measure, F, of {X_{m,n}}, satisfies F ≪ σ_{2}, (ii) the density f (e^{iλ}, e^{iμ}) = |φ (e^{iλ}, e^{iμ})|^{2}, with φ ∈ H^{2} (T^{2}), (iii) [φ]_{Z_{+}^{2}} = H^{2} (T^{2}).

From the above theorem, we obtain the following result of Soltani [9].

Theorem 3.2. A second order stationary random field has a proper QMA representation if and only if it satisfies conditions (3.3), (3.4), and (i)–(iii) of Theorem 3.1.

Since φ ∈ H^{2} (T^{2}), with [φ]_{Z_{+}^{2}} = H^{2} (T^{2}) (H-outer for two variables), implies

that [φ]_{S} = [[φ]_{Z_{+}^{2}}]_{S} = [H^{2} (T^{2})]_{S} = [S], we get that φ is outer. This was originally proved in [2]. A counter example to the converse of the above statement was provided in [8]. In [7], a necessary and sufficient condition was given for the equivalence of the properties for a function to be H-outer and outer. Thus, in Theorem 3.1, we cannot replace the condition (iii) by the requirement that φ be outer.

We note that if g ∈ H^{2} (T^{2}), then g is of S-analytic type. One can ask whether one can improve Lemma 2.4 under the assumption that g ∈ H^{2} (T^{2}). Let us observe that, with h as in (2.5), we have

g (e^{iλ}, e^{iμ}) = h (e^{iλ}) + e^{iμ}h̃ (e^{iλ}, e^{iμ}) ,

where h̃ ∈ H^{2} (T^{2}). If g is outer, then, with H^{2} (T_{λ}) = \overline{sp} \{e^{ijλ} : j ∈ Z_{+}\} in L^{2} (T^{2}, σ_{2}), we get

H^{2} (T_{λ}) ⊂ [S] = [g]_{S} = \overline{sp} \{e^{ijλ}h (e^{iλ})\} ⊕ M,

with M ⊥ H^{2} (T_{λ}). Since h ∈ H^{2} (T_{λ}), we obtain

H^{2} (T_{λ}) = \overline{sp} \{e^{ijλ}h (e^{iλ}) , j ≥ 0\} .

This shows that h is H-outer in T_{λ} or, equivalently, that h is outer in T. In combination with Lemma 2.4 we obtain the following result from [5], with a simple proof.

Theorem 3.3. Let g ∈ H^{2} (T^{2}); then g is outer if and only if h, defined in (2.5) is outer in T, and g is λ-outer.

Remark 3.4. One can derive other results in [5] with simple variations of the above arguments.

References

1. S. A. Chobanyan and N. N. Vakhaniya, The Linear prediction and approximation of weak second order random elements. Prediction Theory and Harmonic Analysis: The Pesi Masani Volume (V. Mandrekar and H. Salehi, eds.), 37–60, North-Holland, Amsterdam, 1983.

2. H. Helson, Lectures on invariant subspaces. Academic Press, New York, 1964.

3. H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables. Acta Math. 99(1958), 165–202.

4. P. Ghatage and V. Mandrekar, On Beurling type invariant subspaces of L^{2}(T^{2}) and their equivalence. J. Operator Theory 20(1988), 83–89.

5. K. Izuchi and Y. Matsugu, Outer functions and invariant subspaces on the torus. Acta Sci. Math. (Szeged) 59(1994), 429–440.

6. G. Kallianpur and V. Mandrekar, Non-deterministic random fields and Wold and Halmos decomposition for commuting isometries. Prediction Theory and Harmonic Analysis: The Pesi Masani Volume (V. Mandrekar and H. Salehi, eds.), 165–190, North-Holland, Amsterdam, 1983.

7. V. Mandrekar, On the validity of Beurling theorems in polydiscs. Proc. Amer. Math. Soc. 103(1988), 145–148.

8. W. Rudin, Invariant subspaces of H^{2} on a torus. J. Funct. Anal. 61(1985), 378–384.

9. A. R. Soltani, Extrapolation and moving average representation for stationary random fields and Beurling’s theorem. Ann. Probab. 12(1984), 120–132.

(Received 3.07.2000)

Author’s addresses:

L. Gawarecki and P. Richard Kettering University Department of Science and Mathematics 1700 W. Third Ave. Flint, MI 48504, U.S.A. E-mail: lgawarec@kettering.edu

prichard@kettering.edu

V. Mandrekar Department of Statistics and Probability Michigan State University Wells Hall, East Lansing, MI 48840, U.S.A. E-mail: mandrekar@stt.msu.edu