Spectral synthesis via moment functions on hypergroups

1 Introduction

In our former paper (see [3]), we studied the following problem: given a commutative hypergroup X, is it true that all exponential polynomials are included in the linear span of all generalized moment functions? A more precise formulation of this problem reads as follows: given an exponential monomial corresponding to the exponential m on the commutative hypergroup X, is it true that the (finite-dimensional) variety of this exponential monomial has a basis consisting of generalized moment functions associated with the exponential m? In [3], we proved that a sufficient condition for this is that all m-sine functions in the given variety form a one-dimensional subspace. Of course, the same holds if all m-sine functions in the variety are zero, as in that case every exponential monomial, as well as every generalized moment function in the given variety, is a constant multiple of m. In addition, it was shown in [3] that this sufficient condition holds on polynomial hypergroups in one variable, for example.

A natural question is if the condition that the m-sine functions in the given variety form a linear space of dimension at most one is necessary. The subject of this paper is to study this problem, to give a negative answer and to support the conjecture that, in fact, every exponential monomial is a linear combination of generalized moment functions in its variety. In particular, we show that if X is a polynomial hypergroup in d variables, then every exponential monomial is the linear combination of generalized moment functions; however, there are exponential monomials, whose variety contains more than one linearly independent sine functions if d > 1 .

A hypergroup is a locally compact Hausdorff space X equipped with an involution and a convolution operation defined on the space of all bounded complex regular measures on X. For the formal definition, historical background and basic facts about hypergroups, we refer to [2]. In this paper, X denotes a commutative hypergroup with identity element o, involution ⋅ ˇ and convolution * . For each function f on X, we define the function f ˇ by f ˇ ⁢ ( x ) = f ⁢ ( x ˇ ) .

Given x in X, we denote the point mass with support the singleton { x } by δ x . The convolution δ x * δ y is a compactly supported probability measure on X, and for each continuous function h : X → ℂ , the integral

will be denoted by h ⁢ ( x * y ) . Given y in X, the function x ↦ h ⁢ ( x * y ) is the translate of h by y.

A set of continuous complex-valued functions on X is called translation invariant if it contains all translates of its elements. A linear translation invariant subspace of all continuous complex-valued functions is called a variety if it is closed with respect to uniform convergence on compact sets. The smallest variety containing the given function h is called the variety of h , and is denoted by τ ⁢ ( h ) . Clearly, it is the intersection of all varieties including h. A continuous complex-valued function is called an exponential polynomial if its variety is finite-dimensional. An exponential polynomial is called an exponential monomial if its variety is indecomposable, that is, it is not the sum of two proper subvarieties. The simplest nonzero exponential polynomial is the one having one-dimensional variety: it consists of all constant multiples of a nonzero continuous function. Clearly, it is an exponential monomial. If we normalize that function by taking 1 at o, then we have the concept of an exponential. Recall that m is an exponential on X if it is a non-identically zero continuous complex-valued function satisfying m ⁢ ( x * y ) = m ⁢ ( x ) ⁢ m ⁢ ( y ) for each x , y in X. Exponential monomials and polynomials on commutative hypergroups have been introduced and characterized in [11, 7, 9]. In the study of exponential polynomials, the basic tool is the modified difference (see [7]). For a given function f in 𝒞 ⁢ ( X ) , y in X and an exponential m : X → ℂ , we define the modified difference

for all x in X. For y 1 , … , y N + 1 in X, we define

for all x in X. The following characterization theorem of exponential monomials is proved in [11, Corollary 2.7].

If f is nonzero, then m is uniquely determined, and f is called an m-exponential monomial, and its degree is the smallest n satisfying the property in Theorem 1. The degree of each exponential function is zero: in fact, every nonzero exponential monomial of degree zero is a constant multiple of an exponential. For exponential monomials of first-degree sine functions, we provide an example: given an exponential m on X, the continuous function s : X → ℂ is called an m-sine function if

holds for each x , y in X. If s is nonzero, then m is uniquely determined, and its degree is 1.

Important examples for exponential polynomials are provided by the moment functions. Let X be a commutative hypergroup, r a positive integer, and for each multi-index α in ℕ r , let f α : X → ℂ be a continuous function such that f α ≠ 0 for | α | = 0 . We say that ( f α ) α ∈ ℕ r is a generalized moment sequence of rank r if

holds whenever x , y are in X (see [4]). It is obvious that the variety of f α is finite-dimensional: in fact, every translate of f α is a linear combination of the finitely many functions f β with β ≤ α , by equation (1.1). The functions in a generalized moment function sequence are called generalized moment functions. In this paper, we shall omit the “generalized” adjective: we simply use the terms moment function sequence and moment function. The order of f α in the above moment function sequence is defined as | α | . It follows that moment functions are exponential polynomials.

The special case of rank-1 moment sequences leads to the following system of functional equations:

for all k = 0 , 1 , … , N and for each x , y in X.

Observe that, in every moment function sequence, the unique function f ( 0 , 0 , … , 0 ) is an exponential on the hypergroup X. In this case, we say that the exponential m = f ( 0 , 0 , … , 0 ) generates the given moment function sequence and that the moment functions in this sequence correspond to m . We may also say that the moment functions in the given moment sequence are m-moment functions. It is also easy to see that, in every moment function sequence, f α with | α | = 1 is an m-sine function on X, where m = f ( 0 , 0 , … , 0 ) .

The main result in [4] is that moment sequences of rank r can be described by using Bell polynomials if the underlying hypergroup is an Abelian group. The point is that, in this case, the situation concerning (1.1) can be reduced to the case where the generating exponential is the identically 1 function and the problem reduces to a problem about polynomials of additive functions. Unfortunately, if X is not a group, then such a reduction is not available, in general.

The following result is important.

This theorem can be formulated in the way that every m-moment function is an m-exponential monomial.

Exponential monomials are the basic building blocks of spectral synthesis. We say that spectral analysis holds for a variety if every nonzero subvariety in the variety contains a nonzero exponential monomial. We say that a variety is synthesizable if all exponential monomials in the variety span a dense subspace. We say that spectral synthesis holds for a variety if every subvariety of it is synthesizable. It is obvious that spectral synthesis for a variety implies spectral analysis for it. If spectral analysis holds for every variety on X, then we say that spectral analysis holds on X . If every variety on X is synthesizable, then we say that spectral synthesis holds on X . Clearly, on every commutative hypergroup, spectral synthesis holds for finite-dimensional varieties.

In the light of these definitions, our basic problem about the relation between exponential monomials and moment functions can be reformulated: is it true that every finite-dimensional variety is spanned by moment functions? If so, then spectral synthesis is rather based on moment functions, not on exponential monomials. Obviously, it is enough to consider indecomposable varieties, and we may even assume that the variety in question is the variety of an exponential monomial. Our result in [3, Theorem 2.1] says that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions – obviously, associated with m. In the subsequent paragraphs, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety.

2 Polynomial hypergroups

In this section, we recall the definition of a class of hypergroups which will be used in the sequel.

The following definition is taken from [2, Chapter 3, Section 3.1, p. 133]. Let d denote a positive integer, and let 𝒫 n denote the set of polynomials in on ℂ d of degree at most n. Finally, let π denote a probability measure on ℂ d , and we assume that the commutative hypergroup X with convolution * , involution ⋅ ˇ and identity o satisfies the following properties: for each x in X, there exists a polynomial Q x on ℂ d such that

1. for each nonnegative integer n, the set X n = { Q x : x ∈ X } is a basis of the linear space 𝒫 n ;

2. we have Q x ⁢ ( 1 , 1 , … , 1 ) = 1 for each x in X;

3. for each x in X, we have Q x ˇ ⁢ ( z ) = Q ¯ x ⁢ ( z ) whenever z is in supp ⁡ π ;

4. for each x , y , w in X, we have

   ∫ ℂ d Q x ⁢ Q y ⁢ Q w ⁢ 𝑑 π ≥ 0 .

In fact, properties (P1), (P2), (P3), (P4) are not independent – for the details, see [2, Proposition 3.1.4]. Property (P4) expresses that the polynomials Q x for x in X are orthogonal with respect to the measure π.

In this case, by property (P1), for each x , y in X, the polynomial Q x ⁢ Q y admits a unique representation

with some complex numbers c ⁢ ( x , y , w ) . Formula (2.1) is called linearization formula, and the numbers c ⁢ ( x , y , w ) are called linearization coefficients.

The hypergroup X with convolution * is called a polynomial hypergroup (in d variables) if there exists a family { Q x : x ∈ X } of polynomials satisfying the above property and

for each x , y in X. In this case, we say that the polynomial hypergroup X is associated with the family of polynomials { Q x : x ∈ X } .

The choice d = 1 is the case of polynomial hypergroups in one variable. In this case, we may suppose that X = ℕ , the set of natural numbers, and the family of polynomials { Q x : x ∈ X } is in fact a sequence of polynomials ( P n ) n ∈ ℕ , where P n is of degree n and we always assume that P 0 = 1 . Suppose that there exists a Borel measure on a finite interval such that the sequence ( P n ) n ∈ ℕ is orthogonal with respect to this measure. In this case, it turns out that the linearization coefficients c ⁢ ( k , l , n ) are zero unless | k - l | ≤ n ≤ k + l . Hence the linearization formula has the form

In this situation, the sequence ( P n ) n ∈ ℕ generates a polynomial hypergroup if and only if the linearization coefficients are nonnegative.

As a particular example, we consider the sequence ( T n ) n ∈ ℕ of Chebyshev polynomials of the first kind defined by the recursion T 0 ⁢ ( x ) = 1 , T 1 ⁢ ( x ) = x and

for n = 1 , 2 , … . In this case, it is easy to check that the linearization coefficients are nonnegative and the resulting hypergroup is the Chebyshev hypergroup. For more about polynomial hypergroups in one variable, see also [2, 8].

Based on [2, Subsection 3.1.3 (iii)], this construction can be generalized for dimension d > 1 . Indeed, if K ( 1 ) and K ( 2 ) are polynomial hypergroups, then K ( 1 ) × K ( 2 ) can be made into a polynomial hypergroup, as well. The collection { Q x ⊗ Q y : ( x , y ) ∈ K ( 1 ) × K ( 2 ) } of polynomials on ℂ d 1 + d 2 defines a hypergroup structure on K ( 1 ) × K ( 2 ) , where { Q x ( 1 ) : x ∈ K ( 1 ) } and { Q y ( 2 ) : y ∈ K ( 2 ) } denote the collections of polynomials on ℂ d 1 and ℂ d 2 defining K ( 1 ) and K ( 2 ) , respectively.

Thus, the notion of Chebyshev hypergroups can also be extended for d ≥ 2 . For example, in case d = 2 , after the above construction, a hypergroup arises that can also be defined with the following recursion. Let T 0 , 0 ⁢ ( x , y ) = 1 , T 1 , 0 ⁢ ( x , y ) = x , T 0 , 1 ⁢ ( x , y ) = y , T 1 , 1 ⁢ ( x , y ) = x ⁢ y , and for each nonnegative integers k , n , let

furthermore, for each positive integers k , n with k + n ≥ 2 ,

Clearly, this is satisfied by T k , n ⁢ ( x , y ) = T k ⁢ ( x ) ⋅ T n ⁢ ( y ) , which leads to a two-dimensional generalization of the Chebyshev hypergroup on the basic set X = ℕ × ℕ .

From the above recursion formulas, we can derive

which provides us the convolution of point masses in X,

whenever k , l , m , n are natural numbers. The involution on X is the identity mapping, and the identity is δ 0 , 0 .

Exponential functions on X are described in the following theorem (see [2, Proposition 3.1.2], [8, Theorem 3.1]).

Using this theorem, it is reasonable to denote the exponential corresponding to the pair ( λ , μ ) in ℂ 2 by M λ , μ .

It follows that the linear space of all M λ , μ -sine functions on X is at most two-dimensional, it is spanned by the two functions ( x , y ) ↦ T x ′ ⁢ ( λ ) ⁢ T y ⁢ ( μ ) and ( x , y ) ↦ T x ⁢ ( λ ) ⁢ T y ′ ⁢ ( μ ) . On the other hand, these two functions are linearly independent. Indeed, assume that

holds for some complex numbers α , β and for all x , y in X. Substituting x = 0 , y = 1 and using T 0 ⁢ ( z ) = 1 and T 1 ⁢ ( z ) = z , we have T 0 ′ ⁢ ( λ ) = 0 and T 1 ′ ⁢ ( μ ) = 1 ; hence we get β = 0 . Interchanging the role of x and y, we obtain α = 0 ; hence the two functions ( x , y ) ↦ T x ′ ⁢ ( λ ) ⁢ T y ⁢ ( μ ) and ( x , y ) ↦ T x ⁢ ( λ ) ⁢ T y ′ ⁢ ( μ ) are linearly independent – they form a basis of the linear space of all M λ , μ -sine functions. We consider the variety τ ⁢ ( φ ) of an M λ , μ -exponential monomial, which is not a linear combination of moment functions in τ ⁢ ( φ ) associated with the exponential M λ , μ . From [3, Theorem 2.1] it follows, that the two functions ( x , y ) ↦ T x ′ ⁢ ( λ ) ⁢ T y ⁢ ( μ ) and ( x , y ) ↦ T x ⁢ ( λ ) ⁢ T y ′ ⁢ ( μ ) must belong to τ ⁢ ( φ ) ; hence the linear space of all M λ , μ -sine functions in τ ⁢ ( φ ) is two-dimensional. We will show that still τ ⁢ ( φ ) is generated by moment functions. This would verify that the sufficient condition given in [3, Theorem 2.1] for that the variety of an exponential monomial is generated by moment functions is not necessary. In fact, we will show that, in general, on every polynomial hypergroup, every finite-dimensional variety is spanned by moment functions. In particular, the variety of each exponential monomial is spanned by moment functions – it follows that every exponential monomial is a linear combination of moment functions. On the other hand, it is easy to see that, on polynomial hypergroups in more than one variable, there are exponential monomials whose variety contains more than one linearly independent sine function.

We note that the formula for S in equation (2.4) can be written in the following form:

which is a special case of the moment functions appearing in Theorem 5 in the next section.

3 A class of moment functions on polynomial hypergroups

Let X be a polynomial hypergroup in d variables associated with the family of polynomials { Q x : x ∈ X } , and we always assume that Q o = 1 , where o is the identity of X. We introduce a special class of exponential monomials on X. Let P be a polynomial in d variables,

where we use multi-index notation. This means that ξ = ( ξ 1 , ξ 2 , … , ξ d ) is in ℂ d , α = ( α 1 , α 2 , … , α d ) is in ℕ d , a α is a complex number and

Then we write

where ∂ = ( ∂ 1 , ∂ 2 , … , ∂ d ) with the obvious meaning of the partial differential operators ∂ i . The differential operator P ⁢ ( ∂ ) is defined on the space of polynomials in d variables in the usual way: given the polynomial Q in the polynomial ring ℂ ⁢ [ z 1 , z 2 , … , z d ] , then

for each ξ in ℂ d . If P is not identically zero, then we always assume that ∑ | α | = N | a α | > 0 , that is, the total degree of the polynomial P is N – in this case, we say that the differential operator P ⁢ ( ∂ ) is of order N.

The following result shows that the functions x ↦ P ⁢ ( ∂ ) ⁢ Q x are linear combinations of moment functions on X.

The moment functions of the form x ↦ ∂ ⁡ Q x ⁢ ( λ ) play an important role in our work: our purpose is to show that, in any variety, the linear combinations of these moment functions span a dense subspace. In other words, spectral synthesis holds on any polynomial hypergroup even when we restrict ourselves to exponential polynomials merely of the form x ↦ P ⁢ ( ∂ ) ⁢ Q x ⁢ ( λ ) . This will be proved in the subsequent paragraphs.

4 The Fourier–Laplace transform

In what follows, we shall use the Fourier–Laplace transform on commutative hypergroups. Here we shortly summarize the basic concepts and results. Let X be a commutative hypergroup, and let 𝒞 ⁢ ( X ) denote the linear space of all complex-valued continuous functions on X. Equipped with the uniform convergence on compact sets, 𝒞 ⁢ ( X ) is a locally convex topological vector space. If, for instance, X is discrete, then 𝒞 ⁢ ( X ) is the linear space of all complex-valued functions on X and the topology on 𝒞 ⁢ ( X ) is the topology of pointwise convergence.

The topological dual space of 𝒞 ⁢ ( X ) can be identified with the space of all compactly supported complex Borel measures on X, denoted by ℳ c ⁢ ( X ) . The identification depends on the Riesz Representation Theorem (see e.g. [5, Theorem 6.19]): every continuous linear functional Λ on 𝒞 ⁢ ( X ) can be represented in the form

whenever f is in 𝒞 ⁢ ( X ) , where μ is a uniquely determined measure in ℳ c ⁢ ( X ) , depending on Λ only. Clearly, if X is discrete, then ℳ c ⁢ ( X ) is the linear space of all finitely supported complex-valued functions on X. For each measure μ in ℳ c ⁢ ( X ) , we define the measure μ ˇ by

whenever f is in 𝒞 ⁢ ( X ) .

The convolution in X induces an algebra structure on ℳ c ⁢ ( X ) in the following manner. Given two measures μ , ν in ℳ c ⁢ ( X ) , their convolution is introduced as the measure μ * ν defined on 𝒞 ⁢ ( X ) by the formula

whenever f is in 𝒞 ⁢ ( X ) . We recall that f ⁢ ( x * y ) stands for the integral

hence the above formula can be written in the more detailed form as

whenever f is in 𝒞 ⁢ ( X ) . As all the measures μ , ν , μ * ν are compactly supported, this integral exists for each continuous function f.

It is easy to check that the linear space ℳ c ⁢ ( X ) is a commutative algebra with the convolution of measures. We call ℳ c ⁢ ( X ) the measure algebra of the hypergroup X. If X is discrete, then it is usually called the hypergroup algebra of X – imitating the terminology used in group theory.

The Fourier–Laplace transform on ℳ c ⁢ ( X ) is defined as follows: for each measure μ in ℳ c ⁢ ( X ) and for each exponential m on X, we let

Then μ ^ : ℰ ⁢ ( X ) → ℂ is a complex-valued function defined on the set ℰ ⁢ ( X ) of all exponentials on X. Clearly, the mapping μ ↦ μ ^ is linear, and it is also an algebra homomorphism, as is shown by the following simple calculation:

which holds for each μ , ν in ℳ c ⁢ ( X ) and for every exponential m on X.

From this property, it follows that the set of all Fourier–Laplace transforms on ℰ ⁢ ( X ) form an algebra, which is called the Fourier algebra of X, denoted by 𝒜 ⁢ ( X ) . In fact, by one of the most important properties of the Fourier–Laplace transform, it follows that the Fourier algebra is isomorphic to the measure algebra, which is expressed by the following theorem (see e.g. [2, Theorem 2.2.4]).

For our purposes, it will be necessary to describe the Fourier algebra of polynomial hypergroups. This will be done in the subsequent paragraphs.

Let X be a polynomial hypergroup in d variables generated by the family { Q x : x ∈ X } of polynomials. We may assume that Q o = 1 , where o is the identity of X. It is known that the function m : X → ℂ is an exponential on X if and only if there exists a λ in ℂ d such that m ⁢ ( x ) = Q x ⁢ ( λ ) holds for each x in X. As λ is obviously uniquely determined by m, hence the set ℰ ⁢ ( X ) of all exponentials on X can be identified with ℂ d . Consequently, the Fourier–Laplace transform of each measure in ℳ c ⁢ ( X ) is a complex-valued function on ℂ d ,

In fact, the integral is a finite sum, which implies that μ ^ is a complex polynomial in d variables. Conversely, let P be any polynomial in the polynomial ring ℂ ⁢ [ z 1 , z 2 , … , z d ] . Then, by definition, P has a representation of the form

with some elements x k in X and complex numbers c k for k = 1 , 2 , … , n . We have

hence

Here μ = ∑ k = 1 n c k ⁢ δ x k is in ℳ c ⁢ ( X ) ; hence we have proved that each polynomial in ℂ ⁢ [ z 1 , z 2 , … , z d ] is in 𝒜 ⁢ ( X ) . In other words, the Fourier algebra of each polynomial hypergroup in d variables is the polynomial ring in d variables.

5 Spectral synthesis via moment functions

We shall use the following basic result (see [6], [8, Theorem 6.9]).

The following result can be found in [8, Theorem 6.10], but for the sake of completeness, we present it together with its proof.