Complex moment problem, truncated
One of the interpolation problems in the complex domain.
Given a doubly indexed finite sequence of complex numbers :
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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
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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
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[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
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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].
References
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Complex moment problem, truncated. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_moment_problem,_truncated&oldid=15610