Having looked at the basic discretization process by itself in the last section, the starting point for this section are now basic difference equations as concrete basic objects, namely
with suitable nonnegative -coefficients and -coefficients. We want to investigate some specific moment problems related to it.
To do so, let us first have a look at the structures which will come in. Like in the previous section, we will always assume .
Definition 5.1 (Continuous structures in use)
Throughout the sequel, we will make use of the multiplication operator and shift-operator, their actions being given by
on suitable definition ranges. We define the linear functional by
and successively for arbitrary the linear functionals on their maximal domains via
provided the integral in (59) exists in all cases.
We are now going to formulate a technical result concerning solutions to (56) which will be a helpful tool to the investigations in the sequel.
Lemma 5.2 (Extension property for continuous solutions)
Let us consider for a fixed number
the basic difference equation
where we assume that , are nonnegative numbers. We require additionally and . Let moreover for two arbitrary but fixed the sets
be given and let the function , as well as the function , be continuous and in particular in agreement with (56) resp. (60) where at most u or v exclusively is allowed to be the zero function in the sense of the Lebesgue integral.
Then u together with v can be extended via (60) to one single positive solution which is in any case continuous on . The action of all on both hand sides of (60) yields
transforming into the moment equation
Proof Rewriting (60) in the form of (56), one recognizes that the extension process from u together with v to f is standard. By conventional analytic arguments, the existence of all moments of f can be concluded. Hence, in particular, we have , where has to be specified since it does not come out of the extension process.
The choice of may in particular be arbitrary since this does not violate the property . By analytic standard arguments of integration, one may conclude that (62) and (63) are for all well defined. These observations practically conclude all steps while verifying the technical Lemma 5.2. □
Remark 5.3 We would like to point out that Lemma 5.2 already provides a plenty of possible solutions to (60). Let, for instance, be a sequence of functions in the sense of Lemma 5.2 and be a sequence of nonnegative numbers where only finitely many of them are different from zero. The linear combination
is then again a solution to (60) in the sense of Lemma 5.2: The function g is in any case continuous in and is in the maximal domain of K. Hence, the application of all the integration functionals () on the equation
leads to the moment equation
Having shed some light on continuous solutions of the basic difference equation (60), let us now look at restrictions of the continuous solutions to special intervals. To do so, let us confine to subsets of the real axis for which the characteristic function is invariant under the shift operator R from (57) respectively its inverse. We have
Theorem 5.4 (Existence of basic layer solutions)
Let f be a continuous positive solution of (60) on . We look at a basic layer Ω which is a subset of the real axis with positive Lebesgue measure having a characteristic function with properties
The application of all the integration functionals () on the equation
again leads to a moment equation of type (63) resp. (66),
where we have used the abbreviation
Proof Let us illustrate the scenario at the example
where a, b, c, d, e fulfill the respective roles of the coefficients specified in the assertions of Lemma 5.2. We choose now Ω such that the conditions of Theorem 5.4 are fulfilled. Hence, we obtain
According to the properties (67), we can replace the expression on the left-hand side of (72) by :
Integration of the last equality yields
Now we see that it is advantageous to use the symmetry property (67). We obtain by substitution
This step can be generalized by the application of all moment generating functionals, yielding
Here, we see that the specific structure of Ω does not influence the moment equality - what all moment equalities of type (76), however, have in common is the fact that Ω has a symmetry property specified by (67).
Rewriting Eq. (76), we finally end up with a non-autonomous difference equation to determine the evaluation of at the specific :
It is now clear what is the basic philosophy behind determining the moment functionals, so it becomes apparent how the proof in the more general situation of Theorem 5.4 works.
Remark 5.5 According to the assertions of Theorem 5.4 resp. Lemma 5.2, we see that the sequence of moment values is indeed a sequence of positive numbers. In particular, the nonautonomous difference equation (77) conserves this property.
A sufficient condition to ensure the nonnegativity of the numbers is provided by choosing the function f in Theorem 5.4 additionally as symmetric. If this condition is imposed, one can derive from all moments a sequence of orthogonal polynomials, being generated through the choice of f resp. Ω. Note, however, that this condition is not necessary.
There is another example for a sufficient condition: Choose the function f in Theorem 5.4 with the additional property , i.e. the function f is assumed as vanishing on the negative real axis. Under this condition, it can be guaranteed that all are positive.
Given now a positive symmetric continuous solution of (71), the question arises of how two moment sequences and may differ when . It may of course happen that already or resp. and . Therefore, the two sequences may develop in a different manner using the generation process through (77). The underlying orthogonal polynomials will also be different.
Let us now look briefly back on the special situation sketched out in proof of Theorem 5.4. Assume that , , , are four continuous positive solutions of (60) on . Having chosen Ω in a proper way, we learn from Eq. (77) that the sequence of all is in the considered case uniquely determined through the specification of the four values , , , .
Assume that , , , are four continuous positive solutions of (60) on . We ask how we can combine these four functions to one positive continuous solution f of (60) such that after the choice of Ω:
where , , , are four arbitrary given but fixed positive numbers. This corresponds to choosing fixed real coefficients α, β, γ, δ such that
Having solved this system of linear equations for determining α, β, γ, δ, one may check the positivity of the arising separately. Once this is verified, all other moments of are then determined through (77) and one can start constructing the underlying orthogonal polynomials.
This observation now directly motivates another aspect of the arising moment problems: Given two different symmetric positive continuous solutions and on two different basic layers and . Under which conditions will the two sequences and be the same? Or in other words: Under which conditions will be the related orthogonal polynomials the same? The following Corollary 5.6 of Theorem 5.4 sheds some more light to a systematic construction process for a rich variety of the demanded solutions:
Corollary 5.6 (Composing and combining basic layer solutions)
Let be a sequence of positive continuous solutions to (60) in the sense of Theorem 5.4. Moreover, let be a sequence of sets all fulfilling property (67) and in addition being partially pairwise coincident resp. partially pairwise disjoint, i.e.
Let in addition the matrix elements of nonnegative numbers be such that finitely many of its entries are different from zero. The function
leads to the formally same moment equation like (63), (66), (69), namely
One might think of dropping the condition of having finitely many entries different from zero. This more general approach can for instance be established by choosing the sequence such that (81) is guaranteed and such that all the projections fulfill in addition the property
The necessary convergence checks arising from the imposed topological structures then have to be tackled separately.
So far, we have considered the situation of piecewise continuous solutions to (60). We would like to point out that the nonautonomous basic difference equation (60) has of course also purely discrete solutions which may stem from suitable projections on the continuous solutions that we have already considered. To do so, we put first together all discrete tools that we need:
Definition 5.7 (Discrete structures in use)
Like in the continuous case, we will make use of the multiplication operator and shift-operator resp. its inverse, their actions being given by
on the considered functions. Given two arbitrary but fixed positive numbers r, s, we are going to define the set
For a suitable function , we define for all linear functionals via a discretized version of the continuous integral, namely
Note that we explicitly require the existence of these expressions for all by speaking of suitable functions f.
The sets from (87) are geometric progressions with a suitable positive part and a suitable negative part. We first would like to point out the conditions that
as direct inspection shows right away. Thus (89) provides a possible discrete analog of (67).
We are now going to relate the introduced structures to solutions of the equation
with suitable nonnegative -coefficients and -coefficients. Here, we assume that the value x is taken from the set (87). We thus look for discrete solutions of (60). The following Theorem 5.8 reveals a plenty of discrete solution structures. Some of the continuous scenarios will similarly appear.
Theorem 5.8 (Discrete solutions of the BDE)
Like in the continuous case, we refer to the basic difference equation (BDE)
where is a fixed number. We assume again here that as well as are non-negative numbers. We require also additionally and .
Under these assertions, there exists to every pair of nonnegative real numbers with precisely one solution to (91) where the pair of positive numbers is assumed as fixed. The solution is fixed through the requirements
Moreover, to each of these and all , there exists .
The action of all on both hand sides of (91) yields
transforming into the moment equation
being formally the same than (63), (66), (69), (83).
be two sequences of positive numbers such that
Let moreover be a sequence of nonnegative numbers which is different from zero in finitely many cases. Let in addition and be two sequences of non-negative numbers in the sense of statement (1) of this theorem. Let for all the functions be defined on . The positive function
fulfills then - in analogy to (63), (66), (69), (83), (94):
Proof (1) Looking back at Eq. (56), we see that a discrete solution is specified completely through application of (56) as soon as the value say at position r and at position −s is provided. In the assertions of Theorem 5.8, we have chosen the pair such that , i.e., the existence of a nontrivial solution is always guaranteed. In particular, it is allowed that we have solutions with vanishing part on the negative side of the lattice resp. solutions with vanishing part on the positive side of the lattice, the scenario of a vanishing part at the same time on the positive lattice and the negative lattice, however, is excluded.
The existence of for all is according to the asymptotic behaviour of which is specified finally by using (56) - the argumentation follows standard results of converging sequences in analysis.
Let us address in particular the evaluation in more detail. To do so, let us concentrate on the right-hand side of the lattice, the proof for the situation on the left-hand side of the lattice being similar: We can show
according to the following identity:
as well as furthermore
Compare now in particular the last expression in (100) with the right-hand side in (98) to get the identity (98) confirmed in total. Ordering and comparing moreover the index structure in the occurring expressions then finally leads to identity (94).
The result in (97) is a consequence of the linearity of all and the assertion of the pairwise disjointness in (95). Note in particular, that we have allowed in the assertions of Theorem 5.8 only finitely many superpositions of lattices in order to avoid more complicated convergence scenarios. □
Remark 5.9 Comparing the continuous scenario and the discrete scenario, we see that a continuous solution of (60) and a corresponding discrete solution of (91) may lead to the same momenta. This means they will in this case also lead to the same type of underlying orthogonal polynomials. In other words, these polynomials have piecewise continuous orthogonality measures on the one hand and purely discrete orthogonality measures on the other hand. According to the construction principles that we have outlined in the continuous and the discrete case, it even becomes clear that the underlying orthogonal polynomials may have orthogonality measures which are mixtures of a piecewise continuous and a purely discrete part. Let us summarize this amazing fact now in the following.
Corollary 5.10 (Mixture of continuous and discrete solutions)
Let , we define
and similarly for ,
Assume that we have chosen a, b, c, d such that for different integer i, j the sets and resp. and are pairwise disjoint.
Assume, moreover, that for two given positive numbers r, s the set constructed according to (87) is not contained in resp. .
We then can look for a solution fulfilling (60), being continuous on according to Theorem 5.4 and discrete on according to Theorem 5.8.
Under these assertions, the functionals
are for all well defined, leading from (60) to the same nonautonomous moment difference equation on which we have ended up in the other cases: