Kyungpook Mathematical Journal 2022; 62(1): 107-118
Published online March 31, 2022
Copyright © Kyungpook Mathematical Journal.
Truncated Multi-index Sequences Have an Interpolating Measure
Hayoung Choi, Seonguk Yoo*
Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : email@example.com
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
e-mail : firstname.lastname@example.org
Received: June 2, 2021; Revised: November 10, 2021; Accepted: November 15, 2021
In this note we observe that any truncated multi-index sequence has an interpolating measure supported in Euclidean space. It is well known that the consistency of a truncated moment sequence is equivalent to the existence of an interpolating measure for the sequence. When the moment matrix of a moment sequence is nonsingular, the sequence is naturally consistent; a proper perturbation to a given moment matrix enables us to confirm the existence of an interpolating measure for the moment sequence. We also illustrate how to find an explicit form of an interpolating measure for some cases.
Keywords: moment problem, interpolating measure, consistency, rank-one decomposition
We first discuss finite sequences of real numbers and then introduce the result of infinite sequences. Let
In a similar way, we consider the
Some properties of
For a motivation of the main result, let us consider the basic Fibonacci sequence. In particular, take the first six moments and write them as a 2-dimensional moment sequence
Due to the Jordan decomposition theorem, every interpolating measure µ has a decomposition,
We conclude this section with another application of the moment problem to the numerical integration. For more details, readers can refer to .
Definition 1.1. A quadrature (or cubature) rule of size
Example 1.2. (Gaussian Quadrature; size
for every polynomial
The solution is
For the sake of a minimal number of nodes, we want
whose solution is obviously
This method seems to provide an economical way to solve a qudrature problem and we will see the main result of this article gives a technique for more general cases, that is, when a signed measure arises in (1.1).
2. The Consistency and Rank-one Decompositions of Moment Matrices
This Section is designed to introduce some background knowledge for dealing with truncated moment sequences.
2.1 The consistency
We are about to define an algebraic set associated to
This is a property of the moment sequence that guarantees the existence of an interpolating measure. Here is a formal result:
Lemma 2.1. ([5, Lemma 2.3]) Let
(i) There exist
In particular, when a positive
2.2 Rank-one decompositions
After rearranging the terms in (2.3) by the sign of densities, we write a measure µ for a consistent
Proposition 2.2. A minimal interpolating measure for a consistent
Many solutions of TRMP for a positive measure depend on finding a positive moment matrix extension of
, which is a row vector corresponding to the monomials in the degree-lexicographic order.
, which is indeed the rank-one moment matrix generated by the measure .
For example, if
3. Main Result
We will verify that any truncated moment matrix turns out to be
Lemma 3.1. () Assume
As a special case of Lemma 3.1, one can easily prove:
Lemma 3.2. Assume
We are ready to introduce a crucial lemma:
Lemma 3.3. A point
Theorem 3.4. Any truncated moment sequence
Theorem 3.5. Any finite sequence has an interpolating measure.
Before we conclude this note, let us discuss how investigate the location of atoms of an interpolating measure. In addition, an algorithmic approach to find an explicit formula of a measure will be presented through a concrete example. Recall that in the presence of a (positive) representing measure µ for a positive
This result provides an evidence that where the atoms of µ lie for a singular
Example 3.6. We illustrate how to find an interpolating measure of the sequence in (3.1). To find an interpolating measure supported in the algebraic variety of
Therefore, we get an interpolating measure
Removing noise from the original data is a challenging problem in many different fields. Moment sequences need to be modified since data obtained from physical experiments and phenomena are often corrupt or incomplete. By Theorem 3.5, one can find an interpolating measure µ for the given data, which is
Example 3.7. Consider a truncated moment sequence
Construct its moment matrix as follows:
It is easy to check that the representing measure is
which is not positive semidefinite. So, arbitrarily small perturbations of a given sequence eject one from the cone of positive semidefinite matrices. As a result, this sequence does not have a representing measure. Instead, one can find interpolating measures for the sequence. Concretely, one of them is
Concluding Remark. According to the main results, we can confirm the existence of an interpolating measure for any finite sequence. A proper moment matrix perturbation enables us to obtain an invertible moment matrix which is consistent. However, invertible matrices may not be useful to find a specific representation of the measure. Rather, it is more advantageous to obtain a moment matrix whose complete solution is known; for example, we may try to make the resulting matrix to be flat and positive semidefinite at the same time. Recall that the Flat Extension Theorem in  says if
Example 3.8. Consider a sequence
An interpolating measure µ for
The authors are indebted to Professor Ilwoo Cho and Professor Raùl Curto for several discussions that led to a better presentation of this note.
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