How to Solve Convolution–Type Functional Equations?

2. Basic function classes

The example considered in the previous section can be generalized. We may consider any variety V in C(G) – it is always the solution space of a system of convolution–type functional equations, namely, of the following one:

$\mu \mathrm{<mo>*</mo>}f\mathrm{<mo>=</mo>01}cm\mu \in V\mathrm{.}$

If V is Mc(G), then the solution space is {0}. If, however, V is a proper ideal, then it is included in a maximal ideal M. If M is a closed maximal ideal, then M is a nonzero subvariety of V, and, by the maximality of M, it can be shown that M is one dimensional. It is very easy to see that in this case M = Mm with some exponential m.

What happens, if every maximal ideal, which includes V is non-closed? Clearly, if a maximal ideal is non-closed, then it is dense, and in this case V includes no exponential: the corresponding system of convolution–type functional equations has no exponential solution. This case is somewhat pathological: on some very large discrete abelian groups there are systems of convolution–type functional equations having no exponential solution. The exact characterization of these groups is given in [6].

Observe, that the exponentials are exactly the eigenfunctions of all translation operators, or more generally, of all convolution operators of the form $f\mapsto \mu \mathrm{<mo>*</mo>}f$ , where µ is in the measure algebra. We can express this property by saying that the intersection of the kernel of all modified difference operators corresponding to m is the space of constant multiples of m. It is quite natural to ask about the kernel of the powers of modified difference operators, i. e. about the solution space of the system

${\Delta }_{m\mathrm{<mo>;</mo>}{y}_1\mathrm{,}{y}_2\mathrm{,}\dots \mathrm{,}{y}_{n+1}}\mathrm{<mo>*</mo>}f\mathrm{<mo>=</mo>0}\text{ (6)}$

for each $y_1\mathrm{,}{y}_2\mathrm{,}\dots \mathrm{,}{y}_{n+1}$  in G, where m is a given exponential and n is a natural number. Clearly, the solution space of this system is $M_m^{n+1}$ . In the case m = 1 the functional equation (6) is the so-called Fréchet equation (see [1,7]), the solutions of which are called generalized polynomials of degree at most n. The case n = 1 is related to the classical Cauchy functional equation

$f\mathrm{<mo\; stretchy="false">(</mo>}x+y\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}+f\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{<mo\; stretchy="false">)</mo>,}\text{ (7)}$

which implies

${\Delta }_{y_1\mathrm{,}{y}_2}\mathrm{<mo>*</mo>}f\mathrm{<mo>=</mo>0}\text{ (8)}$

with the additional property f(0) = 0. These functions are the so-called additive functions – they are actually homogeneous generalized polynomials of first degree. The general solution of (8) is of the form f = a+c, where a is additive and c is a complex number. These functions are called linear functions.

Observe, that the set of all linear functions is exactly the variety $M_1^2$ . Indeed, f is the solution of (8) if and only if it is annihilated by all elements of M₁· M₁, which means

$f\mathrm{<mo\; stretchy="false">(</mo>}x+y_1+y_2\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>}x+y_1\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>}x+y_2\mathrm{<mo\; stretchy="false">)</mo>}+f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0}$

for each x, y1, y2 in G. With x = 0 we get

$f\mathrm{<mo\; stretchy="false">(</mo>}{y}_1+y_2\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>}{y}_1\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>}{y}_2\mathrm{<mo\; stretchy="false">)</mo>}+f\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>0,}$

which can be written as

$f\mathrm{<mo\; stretchy="false">(</mo>}{y}_1+y_2\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}{y}_1\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>}+f\mathrm{<mo\; stretchy="false">(</mo>}{y}_2\mathrm{<mo\; stretchy="false">)</mo>}-f\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo>,}$

that is, f – f(0) is additive.

Going back to (6), its solutions are called generalized m - exponential monomials of degree at most n. By a simple calculation one can verify that f is an m - exponential monomial if and only if it is a generalized polynomial multiplied by m: f = p · m, where p is a generalized polynomial. Of course, the degree of f is equal to the degree of p.

It is obvious, that linear combinations of generalized polynomials are generalized polynomials, but what about the linear combinations of m-exponential monomials with different m’s? What happens if f is annihilated by $M_{m_1}\cdot M_{m_2}$  with different m1 and m2 ? The answer is given by the following theorem:

Theorem 2 The continuous function f : G → C can be written in the form

$f\mathrm{<mo>=</mo>}{p}_1\cdot m_1+p_2\cdot m_2+\cdots +p_k\cdot m_k\text{ (9)}$

where the pj ’s are generalized polynomials and the mj ’s are different exponentials, if and only if f is in $\mathrm{<mo\; stretchy="false">(</mo>}{M}_{m_1}^{n_1+1}\cdot M_{m_2}^{n_2+1}\cdots M_{m_k}^{n_k+1}\mathrm{<mo\; stretchy="false">)</mo>}$ , where $n_1\mathrm{,}{n}_2\mathrm{,}\dots \mathrm{,}{n}_k$  are natural numbers.

Proof. Assume first that f has the given form (9). We have seen above that the annihilator of the functions of the form p · m, where p is a generalized polynomial of degree at most n, is Mn. Using statement 3. in Theorem 1, we get that f is annihilated by $M_{m_1}^{n_1+1}\cap M_{m_2}^{n_2+1}\cap \dots \cap M_{m_k}^{n_k+1}$ , where $n_1\mathrm{,}{n}_2\mathrm{,}\dots \mathrm{,}{n}_k$  are the degrees of the generalized polynomials in (9). As the powers of different maximal ideals are co-prime, and the intersection of co-prime ideals is equal to their product (see e.g. [8]), the necessity part of the theorem follows.

The converse statement follows exactly by the same argument (see also [3]).

Functions f of the form (9) are called generalized exponential polynomials. Here the degree can be defined as the multi-index $\mathrm{<mo\; stretchy="false">(</mo>}{n}_1\mathrm{,}{n}_2\mathrm{,}\dots \mathrm{,}{n}_k\mathrm{<mo\; stretchy="false">)</mo>}$ .

A natural question arises in connection with these function classes we have introduced: why do we use the adjective "generalized"? In fact, the class of generalized polynomials has a subclass, which is more important from the point of view of systems of convolution–type functional equations: a generalized polynomial is called a polynomial, if its variety is finite dimensional. In general, the variety of the function f in C(G) is the smallest variety containing f: it is the intersection of all varieties containing f, and it is denoted by t(f). For instance, every nonzero additive function a is a polynomial of degree 1, as t(a) is the two dimensional space generated by 1 and a. However, if G = Zw, the direct sum of countable many copies of the integers, then there are genralized polynomials on G, which are not polynomials. Roughly speaking, G is the set of all infinite sequences of integers, with only finitely many nonzero terms. A simple example for a generalized polynomial, which is not a polynomial is the following: let ai : G → Z denote the i-th projection of G, defined by

$a_i\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{x}_i$

for every x in G and for each i in R, then clearly, ai is an additive function. We define

$B\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}y\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _i}{a}_i\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}{a}_i\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{<mo\; stretchy="false">)</mo>}$

for every x, y in G. The sum is finite for every x, y in G, and it is obvious that B is additive in both variables. We let

$f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}B\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>,}$

then it is easy to check that

${\Delta }_{y_1\mathrm{,}{y}_2\mathrm{,}{y}_3}\mathrm{<mo>*</mo>}f\mathrm{<mo>=</mo>0}$

for every $y_1\mathrm{,}{y}_2\mathrm{,}{y}_3$  in G, hence f is a generalized polynomial. On the other hand, a simple calculation shows that t(f) is generated by the functions $\mathrm{1,}{a}_i\mathrm{,}f$  for i in R. As the functions ai are linearly independent, hence t(f) is of infinite dimension (see [9]).

Having introduced the concept of "polynomial", we also omit the adjective "generalized" from "generalized exponential monomial" and "generalized exponential polynomial", if the corresponding variety is finite dimensional. It follows that polynomials, exponential monomials and exponential polynomials on G have a nice description, as it is presented in the following result:

Theorem 3 Let G be a commutative topological group.

1. The function f : G → C is a polynomial of degree n if and only if there exist lineraly independent additive functions $a_1\mathrm{,}{a}_2\mathrm{,}\dots \mathrm{,}{a}_k$  and a complex polynomial p : C^k → C such that

$f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}P\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}{a}_1\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}{a}_2\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}\dots \mathrm{,}{a}_k\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}x\in G$

2. The function f : G → C is an exponential monomial if and only if there exists an exponential m : G → C and a polynomial p : G → such that f = p · m.

3. The function f : G → C is an exponential polynomial if and only if there exist different exponentials $m_1\mathrm{,}{m}_2\mathrm{,}\dots \mathrm{,}{m}_k\mathrm{<mo>:</mo>}G\to C$  and polynomials $p_1\mathrm{,}{p}_2\mathrm{,}\dots \mathrm{,}{p}_k\mathrm{<mo>:</mo>}G\to C$  such that $f\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{I\mathrm{<mo>=</mo>1}}^k}{p}_i\cdot m_i$ . If the $p_i$ ’s are nonzero, then this representation of f is unique.

The basic property of exponential polynomials is expressed by the following theorem (see [10, Corollary 11], [3, Theorem 12.31]).

Theorem 4 Let G be a commutative topological group and f : G ® C a continuous function. The variety of f is finite dimensional if and only if f is an exponential polynomial.

Proof. Here we prove the sufficiency – for the proof of the necessity see [10, Corollary 11]. Clearly, it is enough to show that if f is a polynomial, then t(f) is finite dimensional. Indeed, if p is a polynomial and  is an exponential, then a simple calculation shows that j is in t(p) if and only if j · m is in t(p · m). On the other hand, the variety of the exponential polynomial

$f\mathrm{<mo>=</mo>}{p}_1\cdot m_1+p_2\cdot m_2+\cdots +p_k\cdot m_k$

is included in the sum of the varieties $\tau \mathrm{<mo\; stretchy="false">(</mo>}{p}_i\cdot m_i\mathrm{<mo\; stretchy="false">)</mo>}$ for $i\mathrm{<mo>=</mo>1,2,}\dots \mathrm{,}k$ .

On the other hand, it is easy to check that if the polynomial p has the form

$p\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}P\mathrm{<mo\; stretchy="false">(</mo>}{a}_1\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}{a}_2\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}\dots \mathrm{,}{a}_k\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$

where p is a complex polynomial in k variables, and $a_1\mathrm{,}{a}_2\mathrm{,}\dots \mathrm{,}{a}_k$  are additive functions, then t(p) is generated by the finitely many polynomials $x\mapsto {\partial }^{\alpha }P\mathrm{<mo\; stretchy="false">(</mo>}{a}_1\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}{a}_2\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}\dots \mathrm{,}{a}_k\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$ , where a is a multi-index in N^k.

We shall see that the characteristic property of exponential polynomials generating finite dimensional varieties is of utmost importance in solving systems of convolution–type functional equations.

3. Spectral analysis and synthesis

In his fundamental paper [12] in 1947, Laurent Schwartz proved the following theorem:

Theorem 5 Given any continuous complex valued function on the reals, all exponential polynomials in its variety span a dense subspace.

To understand the importance of this result we observe that if the variety of the function f is C(R), then the statement is obvious: by the Stone-Weierstrass theorem even the polynomials form a dense subspace in C(R). The interesting case is the one where the variety of f is not the whole space C(R): in this case f is called a mean periodic function. The above theorem says that every mean periodic function f can uniformly be approximated on compact sets by exponential polynomials, which satisfy every system of convolution–type equations, which is satisfied by f. Roughly speaking, if we know all exponential polynomial solutions of a system of convolution–type equations, then we know all solutions of that system.

It is quite reasonable to ask if this property holds on more general commutative topological groups. If we have a look at the example above, where we presented a generalized polynomial which is not a polynomial, then we can see that the variety of the function f defined by

$f\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}B\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}$

for x in Zw is a counterexample. Indeed, we established that the variety of f is generated by the functions 1, the additive functions ai for i in Z, and the function f itself, which is not a polynomial. The only exponential polynomials in this variety are linear functions, and it is obvious, that limits of linear functions are linear functions as well, hence f cannot be the limit of polynomials, which are in the variety. In the paper [13] the authors showed that if G is a discrete abelian group and there are infinitely many linearly independent additive function on G, then there are varieties on G for which the statement of Schwartz’s theorem does not hold.

In order to dig deeper we introduce some definitions. We always assume that G is a commutative topological group. Our first observation is that if V is a variety on G and a generalized exponential polynomial is in V, then all the exponentials from which it is built up belong to V, as well. However, we mentioned above that there are commutative groups such that some systems of convolutio–type equations do not have exponential solutions. In other words, it may happen that a variety does not include any exponential: of course, in this case the exponential polynomials cannot span a dense subspace. We shall say that spectral analysis holds for a variety V, if every nonzero subvariety of V includes an exponential. We say that spectral analysis holds on the group G, if spectral analysis holds for each nonzero variety on G. This is equivalent to the property that every maximal ideal in the measure algebra is closed. We say that a variety V on G is synthesizable, if the exponential monomials in V span a dense subspace of V. We say that spectral synthesis holds for V, if every subvariety of V is synthesizable. Finally, we say that the group G is synthesizable, or spectral synthesis holds on G, if every variety on G is synthesizable. Clearly, spectral synthesis for a variety implies spectral synthesis for it, but the converse is not true: in [6] the authors proved that on a discrete abelian group spectral analysis holds if and only if the torsion-free rank of the group is less than the continuum, and in [13] it has been proved that on a discrete abelian group spectral synthesis holds if and only if the torsion-free rank of the group is finite.

In the non-discrete case the situation is more complicated. The first natural question is whether Schwartz’s result can be extended for functions in several variables – in other words, does spectral synthesis hold on for n > 1? Somewhat surprisingly, the answer is negative. In fact, in [14] the author presents two counterexamples in R2. In the first case a system of convolution–type equations – consisting of two equations – is given such that the exponential monomials do not span a dense subspace in the solution space. In the second case another system of six convolution–type equations is presented such that the solution space is nontrivial, but the system has no exponential solution. In the light of these negative results the most interesting question arises: how to characterize those commutative topological groups having spectral synthesis?

In our recent work [15] we introduced a method, called localization of ideals in the Fourier algebra of a locally compact abelian group. The main idea is to consider differential oparators on the Fourier algebra, which are polynomials of first order derivations. Given an ideal we say that it is localizable, if it has the following property: if a function is annihilated by all differential operators, which annihilate the ideal, then this function belongs to the ideal, as well. The ideals of the Fourier algebra correspond in a one-to-one way to the ideals of the measure algebra, hence this localizability concept can be applied for the ideals of the measure algebra. Our main result in this respect is that a closed ideal of the measure algebra is localizable if and only if its annihilator variety is synthesizable. This simple criteria for synthesizability leads to a complete characterization of those locally compact abelian groups having spectral synthesis in the following two results (see [15]).

Theorem 6 The compactly generated locally compact abelian group G is synthesizable if and only if it is topologically isomorphic to R^a ×Z^b × C, where a,b are nonnegative integers with a ≤ 1, and C is a compact abelian group.

In the next theorem B denotes the closed subgroup of compact elements in the group G: those elements, which generate a compact subgroup.

Theorem 7 The locally compact abelian group G is synthesizable if and only if G/B is topologically isomorphic to R^a × Z^b × D, where a, b are nonnegative integers with a ≤ 1, and D is a discrete abelian group of finite rank.

Conclusion

Now we can offer a method for solving systems of convolution–type functional equations on locally compact abelian groups. Given the system (4) first we find its exponential solutions: these exponentials m form the spectrum of the system. These are the common roots of the Fourier transforms of the measures m in G. The next step is to find the "multiplicities" of these roots: these are realized by the exponential monomials p · m corresponding to the spectrum, and they form the spectral set of the system. Here p is a polynomial, which can be written in the form

$p\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}P\mathrm{<mo\; stretchy="false">(</mo>}{a}_1\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}{a}_2\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>,}\dots \mathrm{,}{a}_k\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}$

where the additive functions are supposed to be linearly independent, and p is a complex polynomial in k variables. Substituting p · m into the system we obtain

$\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{<mo>*</mo>}pm\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \int }p\mathrm{<mo\; stretchy="false">(</mo>}x-y\mathrm{<mo\; stretchy="false">)</mo>}m\mathrm{<mo\; stretchy="false">(</mo>}x-y\mathrm{<mo\; stretchy="false">)</mo>}d\mu \mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>0}$

for each x in G and for every m in G. To find the polynomial solutions of this system is a purely algebraic job. By the above theorems, if G satisfies the assumptions, then the solution space of the system consists of the limits of convergent sequences formed by the elements of the spectral set. We note that, if G does not satisfy the conditions of the above theorems, still there is a possibility that spectral synthesis holds for the given system – it depends on the localizability of the ideal in question.

For more about the history, classical problems and results related to spectral analysis and synthesis see e.g. [16-23].