Moment Problems on Bounded and Unbounded Domains

�sing approximation results, we characterize the existence of the solution for a two-dimensional moment problem in the �rst quadrant, in terms of quadratic forms, similar to the one-dimensional case. For the bounded domain case, one considers a space of complex analytic functions in a disk and a space of continuous functions on a compact interval. e latter result seems to give sufficient (and necessary) conditions for the existence of a multiplicative solution.


Introduction
Applying the extension Hahn-Banach type results in existence questions concerning the moment problem is a wellknown technique [1][2][3][4][5][6][7][8][9][10][11].One of the most useful results is lemma of the majorizing subspace (see [12,Section 5.1.2]for the proof of the lattice-version of this lemma; see also [13]).It says that if  is a linear positive operator on a subspace  of the ordered vector space , the target space being an order complete vector lattice , and for each    there is      , then  has a linear positive extension     .Another geometric remark is that in the real case, the sublinear functional from the Hahn-Banach theorem can be replaced by a convex one.e theorem remains valid when the convex dominating functional is de�ned on a convex subset  with some qualities with respect to the subspace  (for instance,     ), where  is the relative interior of ).Here we recall an answer published without proof in 1978 [14], without losing convexity, but strongly generalizing the classical result.e �rst detailed proof was published in 1983 [15].e proof of a similar result, in terms of the moment problem, was published in [10].Here we recall the general statement from [14].One of the reasons is that many other results are consequences of this theorem, including Bauer's theorem [13], Namiokas's theorem, and abstract moment problem-results published �rstly in [9].Part of these generalizations of the Hahn-Banach principle are applied in the present work too.roughout this �rst part,  will be a real vector space,  an order-complete vector lattice,     convex subsets,      a concave operator,      a convex operator,    a vector subspace, and      a linear operator.eorem 1. Assume that e following assertions are equivalent: (a) there is a linear extension      of the operator  such that (b) there are  1     convex and  1     concave operators such that for all e minus-sign appears to construct a convex operator in the le-hand side member and a concave operator in the right side.e idea of sandwich theorem on arbitrary convex subsets   is clear.Most of the applications hold for linear positive operators on linear ordered spaces (  + ), when we take    + ,   ,   ,  a suitable convex operator (a vector-valued norm, a sublinear operator), which "measures the continuity" of the extension .One obtains the following result related to the theorem of H. Bauer ([13, Section 5.4]).eorem 2. Let  be a preordered vector space with its positive cone  + ,      a convex operator,    a vector subspace,      a linear positive operator.e following assertions are equivalent: We recall the following approximation lemma on an unbounded bi-dimensional subset.
Lemma 5 (see [7] and [8, e idea of the proof is to add the ∞ point and to apply the Stone-Weierstrass theorem to the subalgebra generated by the functions exp(− 1 −  2 )     + .en one uses for each such exp-function suitable majorizing or minorizing partial sums polynomials.
Note that Lemma 1.4 [8] asserts the density of positive polynomials in ( 1  ()) + , for any closed subset  of a �nite dimensional space,  being a positive regular Borel Mdeterminate measure.e results of this work are generally applications of the theorems stated above.

Moment Problems on Unbounded Subsets
e �rst result of this section concerns the multidimensional moment problem, being a consequence of the Stone Weierstrass theorem and of other usual results on positive polynomials on [ ∞).It is known that in several dimensions there are positive polynomials on   + which cannot be written involving sums of squares, as in the one-dimensional case.However, the approximation results from below seem to show that we can work with limits of tensor products of polynomials of one variable, for which such representation hold in each variable.is leads to characterizations of the existence of the solutions for multidimensional moment problems in terms of quadratic forms (eorem 6(b)).Let  be a Hilbert space,  1   2 two positive commuting selfadjoint operators acting on , with spectrums (  ),   1 2. We introduce the commutative algebra   ( 1   2 ) of self-adjoint operators [5,12], which is also an order-complete vector lattice: We denote by  the space of all continuous functions   [ ∞) 2 We have already seen that these polynomials are dense in  0 ([0 ∞ 2 .Using the above arguments, the assertion (b) says that we have An application of the majorizing subspace lemma [12] is relation leads to the last conclusion (a).us the proof of (b ⇒ (a is �nished.Since the converse is obvious, the theorem is proved. Sometimes it is useful to �nd su�cient or necessary conditions for the existence of the solution in terms of some positive related sequences (assertion (b) of the next theorem).
Since a direct computation and application of the Carleman criterion [18] show that   exp(− is an M-determinate measure, it follows from the same criterion that  ⋅  is Mdeterminate too.us the proof of (a ⇔ (b is �nished.e equivalence (b ⇔ (c follows easily by the aid of the form of positive polynomials on the nonnegative semi-axes.
[1])th the modulus dominated by a polynomial at each point of [ ∞)2, and by  () the elements of the base of polynomials, namely,  () ( 1   2 )   Proof.From the Stone-Weierstrass theorem, we can infer that the space  =  0 (( 1  ⊗  0 (( 2  is dense in the space  0 (( 1  × ( 2 .It follows that any continuous nonnegative function  on the product of spectrums can be uniformly approximated from above with elements of .We extend all the functions involved with zero outside the compacts where they were de�ned, using then Luzin�s theorem.usweobtainpositivecontinuousapproximations with compact support of , de�ned on [0 ∞2.Each such approximation is the limit of a sequence of positive functions from  0 ([0 ∞ ⊗  0 ([0 ∞.Lt  =  1 ⊗  2 be such a function.For each of the separate variables  1   2 , there is a sequence of positive polynomials on the whole nonnegative semi-axes such that   >  ≥ 0 ∀   +    ⟶     ⟶ ∞  = 1 2(13)the convergence being uniform on compact subsets.Because the polynomials involved are positive on the whole interval [0 ∞, from[1]we know their form:     =  2     +    2       = 1 2    + .
leads to the existence of a positive linear extension      of .Using the uniform convergence of the special polynomials on the product of spectrums, we obtain ∑       ≥  for all  ≥     ∑        ∑     ⋅ Γ(  , for all     .(c) for any �nite family (     of real numbers, one has Proof ((b  (a).Denote   (    ,     .One applies eorem 4, (b  (a.Let     and (     in  such that      −            ([ ∞     Application of eorem 4 shows that there exists a linear positive form    *  satisfying the moment conditions and being majorized by   on the positive cone   .Because of the density of   ([ ∞ ⊂  in    , and of the continuity of   with respect to    -norm, there is an extension (a) there exists a unique    ∞  ([0 ∞ such that  ∞     (  (  =   ∀   +  0 ≤  ( ≤ 1 a.e. (18)      being the subspace of polynomials.From the approximation lemma mentioned in the end of the introduction (see [7, Lemma 1.4]), and using (b), we infer that:     ( ∀   ⟹      a.e. If    , consider the following assertions:(a) there is a real linear functional  on  such that     =       +  One observes that for all    + , we have ‖‖ ∞ =  =   .To apply eorem 4, let us consider the following relation and its derivates: Remark 10.For �xed  <  <  = 1, eorem 9 gives an upper bound and a lower bound for the convex set of solutions, both of them being Dirac measures multiplied with a constant.eseboundsare realized if and only if we have   = ±1/1   for some    + .elastresults are applications of eorem 3. Let  be a nonempty set,  an algebra of functions on  and  + the cone of pointwise nonnegative functions on .Let  be an order complete lattice, which is also a commutative algebra.Let (a) there exists a linear positive operator    +   such that     =     ∀                       .Proof.e implication a ⇒ b is obvious.For the converse observe that any element of    is majorized by an element from .Writing this for  too, we get that any    is between two elements  1  ≤  ≤  2 ,     ,  = 1 2. We denote   =        =           .(34) If  ∶    is the linear operator verifying the moment conditions, then (b) says that  is linear and positive on , so that for any  1 ≤ ,  1   we have   1  ≤   .(36)