### Article

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 : hayoung.choi@knu.ac.kr

Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea

e-mail : seyoo@gnu.ac.kr

**Received**: June 2, 2021; **Revised**: November 10, 2021; **Accepted**: November 15, 2021

### Abstract

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

### 1. Introduction

We first discuss finite sequences of real numbers and then introduce the result of infinite sequences. Let

where

In a similar way, we consider the

When

Some properties of _{d}(n)_{d}(n)_{d}(n)_{d}(n)_{d}(n)

where

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, ^{+}^{-}

For ^{+}^{-}

We conclude this section with another application of the moment problem to the numerical integration. For more details, readers can refer to [11].

**Definition 1.1.** A **quadrature** (or **cubature**) rule of size

**Example 1.2.** **(Gaussian Quadrature; size n, precision 2n-1)** We would like to find nodes

for every polynomial

If

The solution is

where

For the sake of a minimal number of nodes, we want ^{2}=1/3

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 _{d}(n)_{d}(n)_{d}(n)_{d}(n)

Given

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

(ii) If

If

For _{d}(n)_{d}(n)_{d}(n)(β)_{d}(n)

In particular, when a positive _{d}(n)_{d}(n)

### 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 _{d}(n)

where

**Proposition 2.2.** A minimal interpolating measure for a consistent _{d}(n)

_{d}(n)_{d}(n)

Since

Many solutions of TRMP for a positive measure depend on finding a positive moment matrix extension of _{d}(n)_{d}(n)

(i)

$v(w):=\left(\begin{array}{c}1\text{\hspace{0.17em}}{w}_{1}\text{}\cdots {w}_{d}{w}_{1}^{2}{w}_{1}{w}_{2}{w}_{1}{w}_{3}\text{}\cdots {w}_{d-1}{w}_{d}{w}_{d}^{2}\text{}\cdots {w}_{1}^{n}\text{}\cdots {w}_{d}^{n}\end{array}\right)$ , which is a row vector corresponding to the monomials${w}^{i}$ in the degree-lexicographic order.(ii)

$P(w):=v{(w)}^{T}v(w)$ , which is indeed the rank-one moment matrix generated by the measure${\delta}_{w}$ .

For example, if

Thus, if _{d}(n)

### 3. Main Result

We will verify that any truncated moment matrix turns out to be

**Lemma 3.1.** ([12]) 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

_{d}(n)

**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 _{d}(n)_{d}(n)

Note that _{2}(1)_{2}(1)

**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 _{2}(1)

for some

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 _{2}(2)

**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 [4] says if _{d}(n)

**Example 3.8.** Consider a sequence

Observe that

An interpolating measure µ for

### Acknowledgments.

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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