Complex moment problem, truncated

One of the interpolation problems in the complex domain.

Given a doubly indexed finite sequence of complex numbers :

with and , the truncated complex moment problem entails finding a positive Borel measure supported in the complex plane such that

is a truncated moment sequence (of order ) and is a representing measure for . The truncated complex moment problem serves as a prototype for several other moment problems to which it is closely related: the full complex moment problem prescribes moments of all orders, i.e., ; the -moment problem (truncated or full) prescribes a closed set which is to contain the support of the representing measure [a26]; and the multi-dimensional moment problem extends each of these problems to measures supported in [a14]; moreover, the -dimensional complex moment problem is equivalent to the -dimensional real moment problem [a8]. All of these problems generalize classical power moment problems on the real line, whose study was initiated by Th.J. Stieltjes (1894), H. Hamburger (1920–1921), F. Hausdorff (1923), and M. Riesz (1923) (cf. also Moment problem and [a1], [a27]).

The truncated complex moment problem is also related to subnormal operator theory [a24], [a29], [a31], polynomial hyponormality [a11], and joint hyponormality [a32], [a33] (cf. also Semi-normal operator). Indeed, A. Atzmon [a2] used subnormal operator theory to solve the full complex moment problem for the disc, and M. Putinar [a18] found a related but different solution to the disc problem based on hyponormal operator theory. More generally, K. Schmüdgen [a26] used an approach based on operator theory and semi-algebraic geometry to obtain the following existence theorem for representing measures [a26] in the multi-dimensional full -moment problem for the case when is compact and semi-algebraic; this result encompasses several previously known special cases (cf. [a4], [a5], [a15], [a17]).

Let denote the multi-shift operator on multi-sequences and let be a finite subset of . Suppose that the semi-algebraic set is compact. Then an -dimensional full (real) moment sequence has a representing measure supported in if and only if the quadratic forms associated with and are positive semi-definite (for every that is a product of distinct ).

For general closed sets , the full -moment problem continues (1998) to defy a complete solution. Hamburger's classical theorem (1920) gives necessary and sufficient conditions for the solvability of the full moment problem on the real line, i.e., : A real sequence with has a representing measure supported in if and only if for each , the Hankel matrix is positive semi-definite (cf. also Nehari extension problem; Synthesis problems). Hamburger's theorem serves as a prototype for much of moment theory, because it provides a concrete criterion closely related to the moments. Nevertheless, when (), positivity alone is not sufficient to imply the existence of a representing measure [a3], [a14], [a25] and a concrete condition for solvability of the full moment problem (including solvability of the full complex moment problem for ) remains unknown (to date, 1999, perhaps the most definitive and comprehensive treatments of the full multi-dimensional -moment problem can be found in [a38], [a39]).

In a different direction, M. Riesz (1923) proved that (as above) has a representing measure supported in a closed set if and only if whenever a polynomial (with complex coefficients) is non-negative on , then . E.K. Haviland (1935, [a16]) subsequently extended this result to the multi-variable full -moment problem. Although Riesz' theorem solves the full moment problem in principle, it is very difficult to verify the Riesz criterion for a particular sequence unless is a half-line (the case studied by Stieltjes), an interval (the case studied by Hausdorff) or, as in Schmüdgen's theorem, when is compact and semi-algebraic. The intractability of the Riesz–Haviland criterion is related to lack of an adequate structure theory for multi-variable polynomials that are non-negative on a given set [a5], [a14], [a20], [a22]; in particular, D. Hilbert (1888) established the existence of a polynomial, non-negative on the real plane, that cannot be represented as a sum of squares of polynomials (cf. [a3], [a14], [a25]).

Because a truncated moment problem is finite in nature, one expects that in cases where a truncated moment problem is solvable, it should be possible to explicitly construct finitely atomic representing measures by elementary methods. (See below for such a construction for the truncated complex moment problem.) From this point of view, the multi-variable truncated -moment problem subsumes the multi-variable quadrature problem of numerical analysis (cf. [a6], [a13], [a23], [a31]). In addition, J. Stochel [a28] has proven that if is a multi-variable full moment sequence, and if for each the truncated sequence has a representing measure supported in a closed set , then some subsequence of converges (in an appropriate weak topology) to a representing measure for with . Thus, a complete solution of the truncated -moment problem would imply a solution to the full -moment problem.

Truncated multi-variable moment problems can be analyzed via the positivity and extension properties of the associated moment matrices [a7], [a8]. For the truncated complex moment problem, one associates to the moment matrix , with rows and columns indexed by , as follows: the entry in row and column is . Thus, if is a representing measure for and (the set of polynomials in of degree at most ), then . Here, denotes the coefficient vector of with respect to the above lexicographic ordering of the monomials in . In particular, it follows that is positive semi-definite and that the support of contains at least points [a8] (cf. [a30]).

It can be proven [a8] that has a rank--atomic (minimal) representing measure if and only if and admits an extension to a moment matrix satisfying .

If admits such a flat extension (i.e. an extension that preserves rank), then there is a relation in (the column space of ). It can be shown [a8], Chap. 5, that then admits unique successive flat (positive) extensions , where is determined by in (). The resulting infinite moment matrix induces a semi-inner product on by . The space

is an ideal in , and is an -dimensional Hilbert space on which the multiplication operator is normal [a8], Chap. 4. The spectrum of (cf. also Spectrum of a matrix) then provides the support for the unique (-atomic) representing measure associated with the flat extension .

To explicitly construct , note that since , there is a linear relation in (or, equivalently, in , since [a12]). The polynomial has distinct complex roots, , which provide the support of , and the densities for , are uniquely determined by the Vandermonde equation

[a8], Chap. 4.

Results of [a10] and [a23] imply that the most general finitely atomic representing measures for correspond to positive, finite-rank moment matrix extensions of . Such an extension exists if and only if admits a positive extension , which in turn admits a flat extension [a10]. Examples for which is required are provided in [a37]. On the other hand, examples in [a3], [a25] imply that a positive, infinite rank need not correspond to any representing measure for .

The preceding results suggest the following flat extension problem [a9], [a10]: under what conditions on does admit a flat extension ? Among the necessary conditions for a flat extension is the condition that be recursively generated, i.e. , imply . Although not every recursively generated positive moment matrix admits a flat extension (or even a representing measure [a10], [a36]), several positive results are known:

i) [a8] If , then admits a rank- atomic representing measure.

ii) [a9] If is recursively generated and if there exist such that in , then admits infinitely many flat extensions, each corresponding to a distinct rank -atomic (minimal) representing measure for .

iii) [a9] If is recursively generated and if in for some , where , then admits a unique flat extension .

The preceding approach can be extended to truncated moment problems in any number of real or complex variables; to do this one defines moment matrices subordinate to lexicographic orderings of the variables [a8]. In the case of one real variable, such moment matrices are the familiar Hankel matrices, and the theory subsumes the truncated moment problems of Stieltjes, Hamburger, and Hausdorff [a7] (cf. also Moment problem).

A refinement of the moment matrix technique also leads to an analogue of Schmüdgen's theorem for minimal representing measures in the truncated -moment problem for semi-algebraic sets. Given , , and a polynomial of degree or , there exists a unique matrix such that (), where ; may be expressed as a linear combination of compressions of .

Let , with or . There exists [a34] a rank--atomic (minimal) representing measure for supported in

if and only if admits a flat extension for which (relative to the uniquely determined flat extension ), .

For additional recent (1999) results on the truncated -moment problem, see [a35], [a36].

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