# The Early History of Moment Problems and Non-Negative Polynomials with Gaps:
Sparse Moment Problems, Sparse Positivstellensätze, and Sparse
Nichtnegativstellensätze from a T-System Point of View

Philipp J. di Dio<EMAIL_ADDRESS>Department of Mathematics and
Statistics, University of Konstanz, Universitätsstraße 10, D-78464 Konstanz,
Germany  Zukunftskolleg, Universtity of Konstanz, Universitätsstraße 10,
D-78464 Konstanz, Germany  eMail<EMAIL_ADDRESS>
###### Abstract

We deal with and investigate sparse univariate Positivstellensätze,
Nichtnegativstellensätze, and solutions to sparse moment problems. The paper
relies heavily on results on T-systems by Karlin in 1963 and by Karlin and
Studden in 1966. We gain complete descriptions of all sparse strictly positive
and sparse non-negative algebraic polynomials on $[a,b]$ with $a\geq 0$ and on
$[0,\infty)$. We extend, simplify, and solve the sparse Hausdorff and
Stieltjes moment problem with these results and the methods of adapted spaces
and T-systems.

###### keywords:

moment problem, Positivstellensatz, Nichtnegativstellensatz, sparse, gap,
T-system

###### MSC:

[2020] Primary 44A60, 14P99 ; Secondary 41A10, 12E10.

††journal: arXiv

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 The Beginning of the Moment Problem
    1. 3.1 The Usual Suspects: Well-known Classical Results without Gaps
    2. 3.2 Early Results with Gaps
    3. 3.3 Finitely Atomic Representing Measures: The Richter Theorem
    4. 3.4 Signed Representing Measures: Boas’ Theorem
  4. 4 T-Systems
    1. 4.1 Definition and Basic Properties of T-Systems
    2. 4.2 Examples of T-systems
    3. 4.3 Non-Negativity, Zeros, and Determinantal Representations of Polynomials in T-Systems
    4. 4.4 ET-Systems
  5. 5 Sparse Positivstellensätze and Nichtnegativstellensätze
    1. 5.1 Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$ for general T-Systems
    2. 5.2 Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$ for Algebraic Polynomials
    3. 5.3 Sparse Positivstellensätze and Nichtnegativstellensätze on $[0,\infty)$
    4. 5.4 Sparse Positivstellensätze and Nichtnegativstellensätze on $\mathds{R}$
  6. 6 Summary

##  1 Introduction

The theory of moments (or the moment problem) has been connected to non-
negative polynomials for a long time and this connection is well-known since
Haviland [16] or even dates back further.

The classical moment problem is the following: Given a closed set
$K\subseteq\mathds{R}^{n}$ and a real sequence $s=(s_{\alpha})_{\alpha\in I}$
with $I\subseteq\mathds{N}_{0}^{n}$. When does there exist a measure $\mu$ on
$K$ such that

$s_{\alpha}=\int_{K}x^{\alpha}\leavevmode\nobreak\ \mathrm{d}\mu(x)$

holds for all $\alpha\in I$?

For more on the theory of moments see e.g. [43, 1, 23, 21, 31, 25, 41] and
references therein.

In modern times the theory of moments and the theory of non-negative
polynomials were revived by [40] and then put to useful applications, see e.g.
[24].

By applications and specially by the need for efficient and fast algorithms
the focus more and more turned the last years (and decades) to sparse systems,
i.e., the index set $I\subsetneq\mathds{N}_{0}^{n}$ is not all
$\mathds{N}_{0}^{n}$ and specially not all $\alpha$ with $|\alpha|\leq d$ for
some $d\in\mathds{N}_{0}^{n}$ are present. It should not be surprising that
these sparse systems were studied theoretically. More surprising is it that
the early results in this field are not well-known or even completely
forgotten. And unfortunately it came recently to our attention that several
known results are being reproved in weaker versions [2].

The main purpose of this article is to review the early results in the theory
of sparse moment problems and to show how important results and especially
sparse Positivstellensätze, sparse Nichtnegativstellensätze, and sparse moment
problems follow from these early results. All results presented here are not
contained in the modern literature about the theory of moments [31, 41], about
real algebraic geometry [3], or about (sparse) polynomial optimization [24,
28].

We hope that this treatment will also be useful for the emerging works of
moment problems and polynomials on curves since these often reduce to the
univariate polynomial case [50].

By the title we only look at early results (and their applications). By
“early” we mean everything up to and including 1966. Everything between 1967
and up to 1991 we consider “modern” and everything after “contemporary”.
Modern and contemporary results are not considered here since they deserve
more space than this article can give. The year 1966 is chosen since in 1966
the extensive research monograph by Samuel Karlin and William J. Studden about
T-systems appeared [23]. This monograph is an extensive follow up of the work
by Karlin in [20]. Both works solve important problems in the theory of
T-systems. The theory of T-systems is the theoretical framework where e.g.
sparse univariate algebraic polynomial systems were investigated in. The year
1991 is chosen since then the first denominator free description of strictly
positive polynomials appeared [40] reviving a large part in real algebraic
geometry.

The article is structured as follows. In the next Section 2 we shortly
introduce the notations in this article. In Section 3 we shortly present the
“usual suspects” (classical results without gaps) and the two first explicit
studies of problems with gaps. We will meet there also Richter’s Theorem and
Boas’ Theorem. In Section 4 we will introduce the theory of T-systems and show
their basic properties with a special emphasis on zeros and non-negativity. By
far the most important part is Section 5 where we look at the results in [20,
23] and apply them to get sparse algebraic Positivstellensätze and
Nichtnegativstellensätze. Additionally, they are used to solve and even extend
the early sparse moment problems from Section 3. In Section 6 we sum up the
results.

All results are presented with proofs as far as possible. Several are
collected from the literature but translated to nowadays mathematical
language. Also some missing steps are filled in and errors are corrected.

##  2 Preliminaries

Let $n\in\mathds{N}$, $K\subseteq\mathds{R}^{n}$ be closed, and
$s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}$ be a real sequence. We say that
$s$ is a $K$-moment sequence if there exists a measure $\mu$ on $K$ such that

$s_{\alpha}=\int_{K}x^{\alpha}\leavevmode\nobreak\ \mathrm{d}\mu(x).$

The measure $\mu$ is called a representing measure. Unless otherwise denoted
as signed measure all measures are positive. The moment sequence $s$ is called
determined if $\mu$ is unique. We call $s$ a truncated moment sequence when
only finitely many $s_{\alpha}$ with $\alpha\in\mathds{N}_{0}$ are known.

For a real sequence $s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}}$ we call the
linear functional $L_{s}:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}$ defined
by $L_{s}(x^{\alpha})=s_{\alpha}$ the Riesz functional. For
$\beta\in\mathds{N}_{0}$ we define
$X^{\beta}s=(s_{\alpha+\beta})_{\alpha\in\mathds{N}_{0}^{n}}$ the shifted
sequence.

For a sequence $s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}$ we define the
Hankel matrix
$\mathcal{H}(s):=(s_{\alpha,\beta})_{\alpha,\beta\in\mathds{N}_{0}^{n}}$. For
$K\subseteq\mathds{R}^{n}$ we set
$\mathrm{Pos}(K):=\\{f\in\mathds{R}[x_{1},\dots,x_{n}]\,|\,f\geq 0\ \text{on}\
K\\}$. For any set $\mathcal{X}$ we denote by $|\mathcal{X}|$ the cardinality
of $\mathcal{X}$.

##  3 The Beginning of the Moment Problem

###  3.1 The Usual Suspects: Well-known Classical Results without Gaps

The first moment problem that was solved is the following.

######  Stieltjes’ Theorem 3.1.1 ([44]).

Let $s=(s_{i})_{i\in\mathds{N}_{0}}$ be a real sequence. The following are
equivalent:

  1. (i)

$s$ is a $[0,\infty)$-moment sequence (Stieltjes moment sequence).

  2. (ii)

$L_{s}(p)\geq 0$ for all $p\in\mathrm{Pos}([0,\infty))$.

  3. (iii)

$L_{s}(p^{2})\geq 0$ and $L_{Xs}(p^{2})=L_{s}(x\cdot p^{2})\geq 0$ for all
$p\in\mathds{R}[x]$.

  4. (iv)

$s$ and $Xs=(s_{i+1})_{i\in\mathds{N}_{0}}$ are positive semidefinite.

  5. (v)

$\mathcal{H}(s)\succeq 0$ and $\mathcal{H}(Xs)\succeq 0$ for all
$d\in\mathds{N}_{0}$.

3.1.1 in the original proof [44] does not use non-negative polynomials.
Stieljes uses continued fractions and introduces new sequences which we
(nowadays) denote by $s$ and $Xs$.

Stieltjes only proves (i) $\Leftrightarrow$ (iv). The implication (i)
$\Leftrightarrow$ (ii) is 3.1.4, (ii) $\Leftrightarrow$ (iii) is the
description of $\mathrm{Pos}([0,\infty))$, and (iv) $\Leftrightarrow$ (v) is a
reformulation of $s$ and $Xs$ being positive semi-definite.

The next moment problem that was solved is the following.

######  Hamburger’s Theorem 3.1.2 ([12, Satz X and Existenstheorem (§8, p.
289)]).

Let $s=(s_{i})_{i\in\mathds{N}_{0}}$ be a real sequence. The following are
equivalent:

  1. (i)

$s$ is a $\mathds{R}$-moment sequence (Hamburger moment sequence or short
moment sequence).

  2. (ii)

$L_{s}(p)\geq 0$ for all $p\in\mathrm{Pos}(\mathds{R})$.

  3. (iii)

$L_{s}(p^{2})\geq 0$ for all $p\in\mathds{R}[x]$.

  4. (iv)

$s$ is positive semidefinite.

  5. (v)

$\mathcal{H}(s)\succeq 0$.

Hamburger proves similar to Stieltjes [44] the equivalence (i)
$\Leftrightarrow$ (iv) via continued fractions. In [12, Satz XIII] Hamburger
solves the full moment problem by approximation with truncated moment
problems. This was later in a slightly more general framework reproved in
[45]. Hamburger needed to assume that the sequence of measures $\mu_{k}$
(which he called “Belegungen” and denoted by $\mathrm{d}\Phi^{(k)}(u)$) to
converge to some measure $\mu$ (condition 2 of [12, Satz XIII]). Hamburgers
additional condition 2 is nowadays replaced by the vague convergence and the
fact that the solution set of representing measures is vaguely compact [41,
Thm. 1.19], i.e., it assures the existence of a $\mu$ as required by Hamburger
in the additional condition 2.

Shortly after Hamburger the moment problem on $[0,1]$ was solved.

######  Hausdorff’s Theorem 3.1.3 ([13, Satz II and III]).

Let $s=(s_{i})_{i\in\mathds{N}_{0}}$ be a real sequence. The following are
equivalent:

  1. (i)

$s$ is a $[0,1]$-moment sequence (Hausdorff moment sequence).

  2. (ii)

$L_{s}(p)\geq 0$ for all $p\in\mathrm{Pos}([0,1])$.

  3. (iii)

$L_{s}(p^{2})\geq 0$, $L_{Xs}(p^{2})\geq 0$, and $L_{(1-X)s}(p^{2})\geq 0$ for
all $p\in\mathds{R}[x]$.

  4. (iv)

$s$, $Xs$, and $(1-X)s$ are positive semidefinite.

  5. (v)

$\mathcal{H}(s)\succeq 0$, $\mathcal{H}(Xs)\succeq 0$, and
$\mathcal{H}((1-X)s)\succeq 0$.

Hausdorff proves the equivalence (i) $\Leftrightarrow$ (iii) via so called
C-sequences. In [48] Toeplitz treats general linear averaging methods. In [13]
Hausdorff uses these. Let the infinite dimensional matrix
$\lambda=(\lambda_{i,j})_{i,j\in\mathds{N}_{0}}$ be row-finite, i.e., for
every row $i$ only finitely many $\lambda_{i,j}$ are non-zero. Then the
averaging method

$A_{i}=\sum_{j\in\mathds{N}_{0}}\lambda_{i,j}a_{j}$

shall be consistent: If $a_{j}\to\alpha$ converges then $A_{i}\to\alpha$
converges to the same limit. Toeplitz proved a necessary and sufficient
condition on $\lambda$ for this property. Hausdorff uses only part of this
property. He calls a matrix $(\lambda_{i,j})_{i,j\in\mathds{N}_{0}}$ with the
property that a convergent sequence $(a_{j})_{j\in\mathds{N}_{0}}$ is mapped
to a convergent sequence $(A_{j})_{j\in\mathds{N}_{0}}$ (the limit does not
need to be preserved) a C-matrix (convergence preserving matrix). Hausdorff
gives the characterization of C-matrices [13, p. 75, conditions (A) – (C)].
Additionally, if $\lambda$ is a C-matrix and a diagonal matrix with diagonal
entries $\lambda_{i,i}=s_{i}$ then $s=(s_{i})_{i\in\mathds{N}_{0}}$ is called
a C-sequence. The equivalence (i) $\Leftrightarrow$ (iii) is then shown by
Hausdorff in the result that a sequence is a $[0,1]$-moment sequence if and
only it is a C-sequence [13, p. 102].

A much simpler approach to solve the $K$-moment problem for any closed
$K\subseteq\mathds{R}^{n}$, $n\in\mathds{N}$, was presented by Haviland. He no
longer used continued fractions but employed the Riesz(–Markov–Kakutani)
representation theorem, i.e., representing a linear functional by integration.

The present Riesz–Markov–Kakutani representation theorem was developed in
several stages. A first version for continuous functions on the unit interval
$[0,1]$ is by F. Riesz [37]. It was extended by Markov to some non-compact
spaces [30] and then by Kakutani to locally compact Hausdorff spaces [19].
Interestingly, it already follows from Daniell’s Representation Theorem [9,
10] with Urysohn’s Lemma [49].

Haviland proved the following.

######  Haviland’s Theorem 3.1.4 ([16, Theorem], see also [15, Theorem] for
$K=\mathds{R}^{n}$).

Let $n\in\mathds{N}$, $K\subseteq\mathds{R}^{n}$ be closed, and
$s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}$ be a real sequence. The
following are equivalent:

  1. (i)

$s$ is a $K$-moment sequence.

  2. (ii)

$L_{s}(p)\geq 0$ for all $p\in\mathrm{Pos}(K)$.

In [15, Theorem] Haviland proves “only” the case $K=\mathds{R}^{n}$ with the
extension method by M. Riesz. In [16, Theorem] this is extended to any closed
$K\subseteq\mathds{R}^{n}$. The idea to do so is attributed by Haviland to
Aurel Wintner [16, p. 164]: “A. Wintner has subsequently suggested that it
should be possible to extend this result [[15, Theorem]] by requiring that the
distribution function [measure] solving the problem have a spectrum [support]
contained in a preassigned set, a result which would show the well-known
criteria for the various standard special momentum problems (Stieltjes,
Herglotz [trigonometric], Hamburger, Hausdorff in one or more dimensions) to
be put particular cases of the general $n$-dimensional momentum problem
mentioned above. The purpose of this note is to carry out this extension.”

In [16] after the general Theorem 3.1.4 Haviland then goes through all the
classical results (Theorems 3.1.1 to 3.1.3, and the Herglotz (trigonometric)
moment problem on the unit circle which we did not included here) and shows
how all these results (i.e., conditions on the sequences) are recovered from
the at this point known representations of non-negative polynomials.

For the Hamburger moment problem Haviland uses

$\mathrm{Pos}(\mathds{R})=\left\\{f^{2}+g^{2}\,\middle|\,f,g\in\mathds{R}[x]\right\\}$

which was already known to Hilbert [17]. For the Stieltjes moment problem he
uses

$\mathrm{Pos}([0,\infty))=\left\\{f_{1}^{2}+f_{2}^{2}+x\cdot(g_{1}^{2}+g_{2}^{2})\,\middle|\,f_{1},f_{2},g_{1},g_{2}\in\mathds{R}[x]\right\\}$
(1)

with the reference to Pólya and Szegö (previous editions of [34, 35]). In [34,
p. 82, ex. 45] the representation (1) is still included while it was already
known before, see [43, p. 6, footnote], that

$\mathrm{Pos}([0,\infty))=\left\\{f^{2}+x\cdot
g^{2}\,\middle|\,f,g\in\mathds{R}[x]\right\\}$ (2)

is sufficient. Also in [41, Prop. 3.2] the representation (1) is used, not the
representation (2).

For the $[-1,1]$-moment problem Haviland uses

$\mathrm{Pos}([-1,1])=\left\\{f^{2}+(1-x^{2})\cdot
g^{2}\,\middle|\,f,g\in\mathds{R}[x]\right\\}.$

For the Hausdorff moment problem he uses that any non-negative polynomial on
$[0,1]$ is a linear combination of $x^{m}\cdot(1-x)^{p-m}$,
$m,p\in\mathds{N}_{0}$, $p\geq m$, with non-negative coefficients. For the
two-dimensional Hausdorff moment problem he uses that any non-negative
polynomial on $[0,1]^{2}$ is a linear combination of $x^{m}\cdot
y^{n}\cdot(1-x)^{p-m}\cdot(1-y)^{q-n}$, $n,m,q,p\in\mathds{N}_{0}$, $p\geq m$,
$q\geq n$, with non-negative coefficients [18]. Hildebrandt and Schoenberg
[18] already solved the moment problem on $[0,1]^{2}$ (and more generally on
$[0,1]^{n}$ for all $n\in\mathds{N}$) getting the same result as Haviland. The
idea of using $\mathrm{Pos}(K)$-descriptions to solve the moment problem was
therefore already used by Hildebrandt and Schoenberg in 1933 [18] before
Haviland uses this in [15] and generalized this in [16] as suggested to him by
Wintner.

With these broader historical remarks we see that of course more people are
connected to Theorem 3.1.4. It might also be appropriate to call Theorem 3.1.4
the _Haviland–Wintner_ or _Haviland–Hildebrand–Schoenberg–Wintner Theorem_.
But as so often, the list of contributors is long (and maybe even longer) and
hence the main contribution (the general proof) is rewarded by calling it just
Haviland Theorem.

As one other solved moment problem of the long list (our list here is far from
complete) is the following.

######  Švenco’s Theorem 3.1.5 ([46]).

Let $s=(s_{i})_{i\in\mathds{N}_{0}}$ be a real sequence. The following are
equivalent:

  1. (i)

$s$ is a $(-\infty,0]\cup[1,\infty)$-moment sequence.

  2. (ii)

$L_{s}(p)\geq 0$ for all $p\in\mathrm{Pos}((-\infty,0]\cup[1,\infty))$.

  3. (iii)

$L_{s}(p^{2})\geq 0$, $L_{(X^{2}-X)s}(p^{2})\geq 0$ for all
$p\in\mathds{R}[x]$.

  4. (iv)

$s$ and $(X^{2}-X)s$ are positive semi-definite.

  5. (v)

$\mathcal{H}(s)\succeq 0$ and $\mathcal{H}((X^{2}-X)s)\succeq 0$.

All moment problems on closed and semi-algebraic sets $K\subseteq\mathds{R}$
follow nowadays easily from 3.1.4 and the fact that any preodering from a
natural description of $K$ is saturated, see e.g. [31, Prop. 2.7.3].

The higher dimensional moment problem is much harder than the one-dimensional
moment problem and in general it is not solved. The reason is that a
description of $\mathrm{Pos}(K)$ is in general unknown. A huge progress in
this field was done by Konrad Schmüdgen in 1991 [40] where he solved the
$K$-moment problem for compact semi-algebraic sets $K\subset\mathds{R}^{n}$,
$n\geq 2$. As a corollary he gained a complete description of strictly
positive $f\in\mathrm{Pos}(K)$. These and subsequence results are discussed
elsewhere [31, 25].

###  3.2 Early Results with Gaps

The early history of moment problems with gaps is very thin. We discuss only
[14] and [4].

Hausdorff just solved 3.1.3 in [13] (submitted 11th February 1920) and in [14]
(submitted 8th September 1920) he treats

$s_{n}=\int_{0}^{1}x^{k_{n}}\leavevmode\nobreak\ \mathrm{d}\mu(x)$

with

$k_{0}=0<k_{1}<k_{2}<\dots<k_{n}<\dots$

for a sequence of real numbers, i.e., not necessarily in $\mathds{N}_{0}$. See
also [43, p. 104]. Since Hausdorff in [14] did not have access to 3.1.4 [16]
or the description of all non-negative linear combinations of
$1,x^{k_{1}},\dots,x^{k_{n}},\dots$ the results in [14] need complicated
formulations and are not very strong. Only with the description of non-
negative linear combinations by Karlin [20] an easy formulation of the result
is possible. We will therefore postpone the exact formulation to 5.2.49,
5.2.51, and 5.2.52 where we present easy proofs using also the theory of
adapted spaces [7, 33, 41].

In [4] Boas investigates the Stieltjes moment problem ($K=[0,\infty)$) with
gaps. Similar to [14] the results are difficult to read and they are
unfortunately incomplete since Boas (like Hausdorff) did not have access to
the description of all non-negative or strictly positive polynomials with gaps
(or more general exponents). We will give the complete solution of the
$[0,\infty)$-moment problem with gaps and more general exponents in 5.3.59.

###  3.3 Finitely Atomic Representing Measures: The Richter Theorem

When working with a truncated moment sequence it is often useful in theory and
applications to replace a representing measure with a finitely atomic measure
without changing the moments. That this is always possible for truncated
moment sequences was first proved in full generality by Richter [36].

######  Richter’s Theorem 3.3.6 ([36, Satz 4]).

Let $n\in\mathds{N}$, $(\mathcal{X},\mathfrak{A})$ be a measurable space, and
$\\{f_{i}\\}_{i=1}^{n}$ be a family of real measurable functions
$f_{i}:\mathcal{X}\to\mathds{R}$. Then for every measure $\mu$ on
$\mathcal{X}$ such that all $f_{i}$ are $\mu$-integrable, i.e.,

$s_{i}:=\int_{\mathcal{X}}f_{i}(x)\leavevmode\nobreak\
\mathrm{d}\mu(x)\quad<\infty$

for all $i=1,\dots,n$, there exists a $K\in\mathds{N}$ with $K\leq n$, points
$x_{1},\dots,x_{K}\in\mathcal{X}$ pairwise different, and
$c_{1},\dots,c_{K}\in(0,\infty)$ such that

$s_{i}=\sum_{j=1}^{K}c_{j}\cdot f_{i}(x_{j})$

holds for all $i=1,\dots,n$.

The history of this result is often still misrepresented in the literature,
even after K. Schmüdgen and the present author compared the different
contributions and publication dates in detail in [11]. With these historical
remarks it also is appropriate to call Theorem 3.3.6 the
_Richter–Rogosinski–Rosenbloom Theorem_ [36, 38, 39]. Every other result
before or after [36] is only a special case and can easily be recovered from
3.3.6, especially [6].

Since 3.3.6 only needs a family of finitely many measurable functions it also
includes all cases of gaps in the truncated moment theory.

###  3.4 Signed Representing Measures: Boas’ Theorem

In the theory of moments almost exclusively the representation by non-negative
measures is treated. The reason is the following.

######  Boas’ Theorem 3.4.7 ([5] or e.g. [43, p. 103, Thm. 3.11]).

Let $s=(s_{i})_{i\in\mathds{N}_{0}}$ be a real sequence. Then there exist
infinitely many signed measures $\mu$ on $\mathds{R}$ and infinitely many
signed measures $\nu$ on $[0,\infty)$ such that

$s_{i}=\int_{\mathds{R}}x^{i}\leavevmode\nobreak\
\mathrm{d}\mu(x)=\int_{0}^{\infty}x^{i}\leavevmode\nobreak\ \mathrm{d}\nu(x)$

holds for all $i\in\mathds{N}_{0}$.

3.4.7 also holds in the $n$-dimensional case on $\mathds{R}^{n}$ and
$[0,\infty)^{n}$ for any $n\in\mathds{N}$. See [43, p. 104] for an extension
which kinds of measures can be chosen.

3.4.7 also covers the case with gaps. If any gaps in the real sequence $s$ are
present then fill them with any real number you like.

##  4 T-Systems

We have seen the early attempts to deal with gaps in the moment problems. A
sufficient solution was at these times not possible. Only with the
introduction of so called T-systems and their rigorous investigation
significant progress and finally complete solutions were possible. For more on
the early development and history of T-systems see [22, 20, 23, 21].

###  4.1 Definition and Basic Properties of T-Systems

######  Definition 4.1.8.

Let $n\in\mathds{N}$, $\mathcal{X}$ be a set with $|\mathcal{X}|\geq n+1$, and
let $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a family of real functions
$f_{i}:\mathcal{X}\to\mathds{R}$. We call any linear combination

$f=\sum_{i=0}^{n}a_{i}\cdot f_{i}\quad\in\mathrm{lin}\,\mathcal{F}$

with $a_{1},\dots,a_{n}\in\mathds{R}$ a _polynomial_. The family $\mathcal{F}$
on $\mathcal{X}$ is called a _Tchebycheff system_ (_T-system_) _of order $n$
on $\mathcal{X}$_ if any polynomial $f\in\mathrm{lin}\,\mathcal{F}$ with
$\sum_{i=0}^{n}a_{i}^{2}>0$ has at most $n$ zeros on $\mathcal{X}$.

If $\mathcal{X}$ is a topological space and $\mathcal{F}$ is a family of
continuous functions then we call $\mathcal{F}$ a _continuous T-system_. If
additionally $\mathcal{X}$ is the unit circle then we call $\mathcal{F}$ a
_periodic T-system_.

######  Corollary 4.1.9.

Let $n\in\mathds{N}$ and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a T-system of
order $n$ on some $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$. Let
$\mathcal{Y}\subset\mathcal{X}$ with $|\mathcal{Y}|\geq n+1$. Then
$\mathcal{G}:=\\{f_{i}|_{\mathcal{Y}}\\}_{i=0}^{n}$ is a T-system of order $n$
on $\mathcal{Y}$.

###### Proof.

Let $f\in\mathrm{lin}\,\mathcal{F}$. Then $f$ has at most $n$ zeros in
$\mathcal{X}$ and hence $f|_{\mathcal{Y}}$ has at most $n$ zeros in
$\mathcal{Y}\subset\mathcal{X}$. Since for any $g\in\mathrm{lin}\,\mathcal{G}$
there is a $f\in\mathrm{lin}\,\mathcal{F}$ such that $g=f|_{\mathcal{Y}}$ we
have the assertion. ∎

The set $\mathcal{X}$ does not require any structure or property except
$|\mathcal{X}|\geq n+1$.

In the theory of T-systems we often deal with one special matrix. We use the
following abbreviation.

######  Definition 4.1.10.

Let $n\in\mathds{N}$, $\\{f_{i}\\}_{i=0}^{n}$ be a family of real functions on
a set $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$. We define the matrix

$\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}:=\begin{pmatrix}f_{0}(x_{0})&f_{1}(x_{0})&\dots&f_{n}(x_{0})\\\
f_{0}(x_{1})&f_{1}(x_{1})&\dots&f_{n}(x_{1})\\\ \vdots&\vdots&&\vdots\\\
f_{0}(x_{n})&f_{1}(x_{n})&\dots&f_{n}(x_{n})\end{pmatrix}=(f_{i}(x_{j}))_{i,j=0}^{n}$

for any $x_{0},\dots,x_{n}\in\mathcal{X}$.

######  Lemma 4.1.11 (see e.g. [21, p. 31]).

Let $n\in\mathds{N}$, $\mathcal{X}$ be a set with $|\mathcal{X}|\geq n+1$, and
$\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a family of real functions
$f_{i}:\mathcal{X}\to\mathds{R}$. The following are equivalent:

  1. (i)

$\mathcal{F}$ is a T-system of order $n$ on $\mathcal{X}$.

  2. (ii)

The determinant

$\det\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}$

does not vanish for any pairwise distinct points
$x_{0},\dots,x_{n}\in\mathcal{X}$.

###### Proof.

(i) $\Rightarrow$ (ii): Let $x_{0},\dots,x_{n}\in\mathcal{X}$ be pairwise
distinct. Since $\mathcal{F}$ is a T-system we have that any non-trivial
polynomial $f$ has at most $n$ zeros, i.e., the matrix

$\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}$

has trivial kernel and hence its determinant is non-zero. Since
$x_{0},\dots,x_{n}\in\mathcal{X}$ are arbitrary pairwise distinct we have
(ii).

(ii) $\Rightarrow$ (i): Assume there is a polynomial $f$ with
$\sum_{i=0}^{n}a_{i}^{2}>0$ which has the $n+1$ pairwise distinct zeros
$z_{0},\dots,z_{n}\in\mathcal{X}$. Then the matrix

$Z=\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
z_{0}&z_{1}&\dots&z_{n}\end{pmatrix}$

has non-trivial kernel since $0\neq(a_{0},a_{1},\dots,a_{n})^{T}\in\ker Z$ and
hence $\det Z=0$ in contradiction to (ii). ∎

######  Corollary 4.1.12.

Let $n\in\mathds{N}$, and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a T-system of
order $n$ on some $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq
n+1$. The following hold:

  1. (i)

The functions $f_{0},\dots,f_{n}$ are linearly independent over $\mathcal{X}$.

  2. (ii)

For any $f=\sum_{i=0}^{n}a_{i}\cdot f_{i}\in\mathrm{lin}\,\mathcal{F}$ the
coefficients $a_{i}$ are unique.

###### Proof.

Follows immediately from 4.1.11 (i) $\Rightarrow$ (ii). ∎

We even have the following.

######  Theorem 4.1.13 (see e.g. [21, p. 33]).

Let $n\in\mathds{N}$, $\mathcal{F}$ be a T-system on some set $\mathcal{X}$
with $|\mathcal{X}|\geq n+1$, and let $x_{0},\dots,x_{n}\in\mathcal{X}$ be
pairwise different points. The following hold:

  1. (i)

Any $f\in\mathrm{lin}\,\mathcal{F}$ is uniquely determined by its values
$f(x_{0}),\dots,f(x_{n})$.

  2. (ii)

For any $y_{0},\dots,y_{n}\in\mathds{R}$ there exists a unique
$f\in\mathrm{lin}\,\mathcal{F}$ with $f(x_{i})=y_{i}$ for all $i=0,\dots,n$.

###### Proof.

(i): Since $f\in\mathrm{lin}\,\mathcal{F}$ we have $f=\sum_{i=0}^{n}a_{i}\cdot
f_{i}$. Let $x_{1},\dots,x_{n}\in\mathcal{F}$ be pairwise distinct. Then by
4.1.11 (i) $\Rightarrow$ (ii) we have that

$\begin{pmatrix}f(x_{0})\\\ \vdots\\\
f(x_{n})\end{pmatrix}=\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}\cdot\begin{pmatrix}\alpha_{0}\\\
\vdots\\\ \alpha_{n}\end{pmatrix}$

has the unique solution $\alpha_{0}=a_{0}$, …, $\alpha_{n}=a_{n}$.

(ii): By the same argument as in (i) the system

$\begin{pmatrix}y_{0}\\\ \vdots\\\
y_{n}\end{pmatrix}=\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}\cdot\begin{pmatrix}\alpha_{0}\\\
\vdots\\\ \alpha_{n}\end{pmatrix}$

has the unique solution $\alpha_{0}=a_{0}$, …, $\alpha_{n}=a_{n}$. ∎

So far we imposed no structure on $\mathcal{X}$. We now impose structure on
$\mathcal{X}$. The following structural result was proved in [27] for compact
subsets $\mathcal{X}$ of $\mathds{R}^{n}$ and for arbitrary compact sets
$\mathcal{X}$ in [42, 8].

######  Theorem 4.1.14 ([27, Thm. 2], [42], [8, Thm. 8 and Cor.]).

Let $n\in\mathds{N}$ and $\mathcal{F}$ be a continuous T-system of order $n$
on a topological space $\mathcal{X}$. If $\mathcal{X}$ is a compact metrizable
space then $\mathcal{X}$ can be homeomorphically embedded in the unit circle
$\\{(x,y)\in\mathds{R}^{2}\,|\,x^{2}+y^{2}=1\\}$.

######  Corollary 4.1.15 ([8, Thm. 8]).

The order $n$ of a periodic T-system is even.

###### Proof.

Let $\varphi:[0,2\pi]\to S=\\{(x,y)\in\mathds{R}^{2}\,|\,x^{2}+y^{2}\\}$ with
$\varphi(\alpha)=(\sin\alpha,\cos\alpha)$ and
$\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a periodic T-system. Then the $f_{i}$
are continuous and hence also

$\det\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
t_{0}&t_{1}&\dots&t_{n}\end{pmatrix}$

is continuous in $t_{0},\dots,t_{n}\in S$. If $\mathcal{F}$ is a T-system we
have that

$d(\alpha):=\det\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
\varphi(\alpha)&\varphi(\alpha+2\pi/(n+1))&\dots&\varphi(\alpha+2n\pi/(n+1))\end{pmatrix}$

in non-zero for all $\alpha\in[0,2\pi]$ and never changes singes. If $n$ is
odd then $d(0)=-d(2\pi/(n+1))$ which is a contradiction. Hence, $n$ must be
even. ∎

###  4.2 Examples of T-systems

######  Examples 4.2.16 (algebraic polynomials, see e.g. [23, 21]).

  1. (a)

Let $n\in\mathds{N}$ and $\mathcal{X}\subseteq\mathds{R}$ with
$|\mathcal{X}|\geq n+1$. Then the family $\mathcal{F}=\\{x^{i}\\}_{i=0}^{n}$
is a T-system. This follows immediately from the Vandermonde determinant

$\det\begin{pmatrix}1&x&\dots&x^{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}=\prod_{0\leq i<j\leq n}(x_{j}-x_{i})$ (3)

for any $x_{0},\dots,x_{n}\in\mathcal{X}$.

Note that we abuse the notation for the algebraic polynomial cases. The
functions $f_{0},\dots,f_{n}$ should not be denoted by $x^{i}$ but by

$\cdot^{i}:\mathds{R}\to\mathds{R},\ x\mapsto x^{i}.$

However, then we have the notation

$\begin{pmatrix}\cdot^{0}&\cdot^{1}&\dots&\cdot^{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}\qquad\text{or more
general}\qquad\begin{pmatrix}\cdot^{\alpha_{0}}&\cdot^{\alpha_{1}}&\dots&\cdot^{\alpha_{n}}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}$

which seems hard to read. For convenience we will therefore abuse the notation
and use $x^{i}$ and (3).

  2. (b)

Let $n\in\mathds{N}$, $\mathcal{X}\subseteq[0,\infty)$ with $|\mathcal{X}|\geq
n+1$, and $\alpha_{0}=0<\alpha_{1}<\dots<\alpha_{n}$ be real numbers. Then
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$ is a T-system of order $n$ on
$\mathcal{X}$.

  3. (c)

Let $n\in\mathds{N}$, $\mathcal{X}\subseteq(0,\infty)$ with $|\mathcal{X}|\geq
n+1$, and $0<\alpha_{0}<\dots<\alpha_{n}$ be real numbers. Then
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$ is a T-system of order $n$ on
$\mathcal{X}$. $\circ$

######  Example 4.2.17 (see e.g. [21, p. 38]).

Let $n\in\mathds{N}$ and $\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$ be reals.
Then

$\mathcal{F}=\left\\{e^{\alpha_{0}x},e^{\alpha_{1}x},\dots,e^{\alpha_{n}x}\right\\}$

is a T-system on any $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq
n+1$. $\circ$

######  Example 4.2.18 (see e.g. [21, p. 37-38]).

Let $n\in\mathds{N}$ and $\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$ be reals.
Then

$\mathcal{F}=\left\\{\frac{1}{x+\alpha_{0}},\frac{1}{x+\alpha_{1}},\dots,\frac{1}{x+\alpha_{n}}\right\\}$

is a continuous T-system on any $[a,b]$ or $[a,\infty)$ with
$-\alpha_{0}<a<b$. $\circ$

######  Example 4.2.19 (see e.g. [21, p. 38]).

Let $n\in\mathds{N}$ and let $f\in C^{n}(\mathcal{X})$ with
$\mathcal{X}=[a,b]$, $a<b$, and $f^{(n)}>0$ on $\mathcal{X}$. Then

$\mathcal{F}=\\{1,x,x^{2},\dots,x^{n-1},f\\}$

is a continuous T-system of order $n$ on $\mathcal{X}=[a,b]$. We can also
allow $\mathcal{X}=(a,b)$, $[a,\infty)$, $(-\infty,b)$, …. $\circ$

######  Example 4.2.20 (see e.g. [23, p. 10]).

Let $n\in\mathds{N}$ and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a (continuous)
T-systems on $\mathcal{X}\subseteq\mathds{R}$ with $|\mathcal{X}|\geq n+1$.
Then for any (continuous) function $r:\mathcal{X}\to(0,\infty)$ the family
$\\{r\cdot f_{i}\\}_{i=0}^{n}$ is a (continuous) T-system. $\circ$

######  Example 4.2.21.

Let $n\in\mathds{N}$, $\\{f_{i}\\}_{i=0}^{n}$ be a (continuous) T-system of
order $n$ on $\mathcal{X}\subseteq\mathds{R}$ and let
$g:\mathcal{Y}\subseteq\mathds{R}\to\mathcal{X}$ be a strictly increasing
(continuous) function. Then $\\{f_{i}\circ g\\}_{i=0}^{n}$ is a (continuous)
T-systems of order $n$ on $\mathcal{Y}$. $\circ$

###  4.3 Non-Negativity, Zeros, and Determinantal Representations of
Polynomials in T-Systems

######  Theorem 4.3.22 (see e.g. [23, p. 20] or [21, p. 33]).

Let $n\in\mathds{N}$, $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a T-system on
some set $\mathcal{X}$ with $|\mathcal{X}|\geq n+1$,
$x_{1},\dots,x_{n}\in\mathcal{X}$ be $n$ distinct points, and let
$f\in\mathrm{lin}\,\mathcal{F}$ be a polynomial. The following are equivalent:

  1. (i)

$f(x_{i})=0$ holds for all $i=1,\dots,n$.

  2. (ii)

There exists a constant $c\in\mathds{R}$ such that

$f(x)=c\cdot\det\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x&x_{1}&\dots&x_{n}\end{pmatrix}.$

###### Proof.

(ii) $\Rightarrow$ (i): Clear.

(i) $\Rightarrow$ (ii): If $f=0$ then $c=0$ so the assertion holds. If $f\neq
0$ then there is a $x_{0}\in\mathcal{X}\setminus\\{x_{1},\dots,x_{n}\\}$ such
that $f(x)\neq 0$. Then also the determinant in (ii) is non-zero and we can
choose $c$ such that both $f$ and the scaled determinant coincide also in
$x_{0}$. By 4.1.12 a polynomial is uniquely determined by $x_{0},\dots,x_{n}$
which shows that (ii) is one and hence the only possible polynomial which
fulfills (i). ∎

So far we treated general T-systems. For further properties we go to
continuous T-systems.

######  Definition 4.3.23.

Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system on
$\mathcal{X}\subseteq\mathds{R}$ an interval, $f\in\mathrm{lin}\,\mathcal{F}$,
and let $x_{0}$ be a zero of $f$. Then $x_{0}\in\mathrm{int}\,\mathcal{X}$ is
called a _non-nodal_ zero if $f$ does not change sign at $x_{0}$. Otherwise
the zero $x_{0}$ is called _nodal_ , i.e., either $f$ changes signs at $x_{0}$
or $x_{0}$ is a boundary point of $\mathcal{X}$.

The following result bounds the number of nodal and non-nodal zeros.

######  Theorem 4.3.24 (see [22] or e.g. [21, p. 34, Thm. 1.1]).

Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system of order $n$ on
$\mathcal{X}=[a,b]$ with $a<b$.

If $f\in\mathrm{lin}\,\mathcal{F}$ has $k\in\mathds{N}_{0}$ non-nodal zeros
and $l\in\mathds{N}_{0}$ nodal zeros in $\mathcal{X}$ then $2k+l\leq n$.

The proof is adapted from [21, Thm. 1.1].

###### Proof.

$\mathcal{X}=[a,b]$ and $k=0$: If $f\in\mathrm{lin}\,\mathcal{F}$ has $l$
zeros then $n\geq l$ by 4.1.8.

$\mathcal{X}=[a,b]$ and $k\geq 1$: Let $x_{1},\dots,x_{n}\in\mathcal{X}$ be
the zeros of $f$. Set

$M_{i}:=\max_{x_{i-1}\leq x\leq x_{i}}|f(x)|$

for $i=1,\dots,k+l+1$ with $t_{0}=a$ and $x_{k+l+1}=b$. Additionally, set

$m:=\frac{1}{2}\min_{i=1,\dots,k+l+1}M_{i}>0.$

We construct a polynomial $g_{1}\in\mathrm{lin}\,\mathcal{F}$ such that

  1. (a)

$g_{1}$ has the values $m$ at the non-nodal zeros $x_{i}$ of $f$ with $f\geq
0$ in a neighborhood of $x_{i}$,

  2. (b)

$g_{1}$ has the values $-m$ at the non-nodal zeros $x_{i}$ of $f$ with $f\leq
0$ in a neighborhood of $x_{i}$, and

  3. (c)

$g_{1}$ vanishes at all nodal zeros $x_{i}$.

After renumbering the $x_{i}$’s we can assume $x_{1},\dots,x_{k_{1}}$ fulfill
(a), $x_{k_{1}+1},\dots,x_{k_{1}+k_{2}}$ fulfill (b), and
$x_{k_{1}+k_{2}+1},\dots,x_{k_{1}+k_{2}+l}$ fulfill (c) with $k_{1}+k_{2}=k$.
By 4.1.8 we have $k+l\leq n$ and hence by 4.1.11 we have that

$\begin{pmatrix}m\\\ \vdots\\\ m\\\ -m\\\ \vdots\\\ -m\\\ 0\\\ \vdots\\\
0\end{pmatrix}=\begin{pmatrix}f_{0}(x_{1})&f_{1}(x_{1})&\dots&f_{n}(x_{1})\\\
\vdots&\vdots&&\vdots\\\
f_{0}(x_{k+l})&f_{1}(x_{k+l})&\dots&f_{n}(x_{k+l})\end{pmatrix}\cdot\begin{pmatrix}\beta_{0}\\\
\vdots\\\ \beta_{n}\end{pmatrix}$

has at least one solution, say $\beta_{0}=b_{0}$, …, $\beta_{n}=b_{n}$. Then
$g_{1}=\sum_{i=0}^{n}b_{i}\cdot f_{i}\in\mathrm{lin}\,\mathcal{F}$ fulfills
(a) to (c).

Set

$\rho:=\frac{m}{2\cdot\|g_{1}\|_{\infty}}$

and let $g_{2}:=f-g_{1}$.

We show that to each non-nodal zero $x_{i}$ of $f$ there correspond two zeros
of $g_{2}$: Let $x_{i}$ be a non-nodal zero of $f$ with $f\geq 0$ in a
neighborhood of $x_{i}$, say. We can find a point $y_{i}\in(x_{i-1},x_{i})$
and a point $y_{i+1}\in(x_{i},x_{i+1})$ such that

$f(y_{i})=M_{i}>m\qquad\text{and}\qquad f(y_{i+1})=M_{i+1}>m.$

Therefore, $g_{2}(y_{i})>0$ and $g_{2}(y_{i+1})>0$. Since
$g_{2}(x_{i})=-\rho\cdot m<0$ if follows that $g_{2}$ as a zero both in
$(y_{i},x_{i})$ and $(x_{i},y_{i+1})$.

Additionally, $g_{2}$ also vanishes at all nodal zeros of $f$ and so has at
least $2k+l$ distinct zeros. Therefore, by 4.1.8 we have $2k+l\leq n$. ∎

######  Corollary 4.3.25.

4.3.24 also holds for sets $\mathcal{X}\subseteq\mathds{R}$ of the form

  1. (i)

$\mathcal{X}=(a,b)$, $[a,b)$, $(a,b]$ with $a<b$,

  2. (ii)

$\mathcal{X}=(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$,

  3. (iii)

$\mathcal{X}=\\{x_{1},\dots,x_{k}\\}\subseteq\mathds{R}$ with $k\geq n+1$ and
$x_{1}<\dots<x_{k}$, and

  4. (iv)

countable unions of (i) to (iii).

###### Proof.

$\mathcal{X}=[0,\infty)$: Let $0\leq x_{1}<\dots<x_{k}$ be the zeros of $f$ in
$[0,\infty)$. Since every T-system on $[0,\infty)$ is also a T-system on
$[0,b]$ for any $b>0$ the claim follows from 4.3.24 with $b=x_{k}+1$.

For the other assertions adapt (if necessary) the proof of 4.3.24. ∎

######  Definition 4.3.26.

Let $x\in[a,b]$ with $a\leq b$. We define the _index_ $\varepsilon(x)$ by

$\varepsilon(x):=\begin{cases}2&\text{if}\ x\in(a,b),\\\ 1&\text{if}\ x=a\
\text{or}\ b.\end{cases}$

The same definition holds for sets $\mathcal{X}$ as in 4.3.25.

######  Definition 4.3.27.

Let $n\in\mathds{N}$ and $\mathcal{F}$ be a T-system of order $n$ on some set
$\mathcal{X}$. We define

$\displaystyle(\mathrm{lin}\,\mathcal{F})^{e}$
$\displaystyle:=\left\\{\sum_{i=0}^{n}a_{i}\cdot
f_{i}\,\middle|\,\sum_{i=0}^{n}a_{i}^{2}=1\right\\},$
$\displaystyle(\mathrm{lin}\,\mathcal{F})_{+}$
$\displaystyle:=\left\\{f\in\mathcal{F}\,\middle|\,f\geq 0\ \text{on}\
\mathcal{X}\right\\},$ and $\displaystyle(\mathrm{lin}\,\mathcal{F})_{+}^{e}$
$\displaystyle:=(\mathrm{lin}\,\mathcal{F})^{e}\cap(\mathrm{lin}\,\mathcal{F})_{+}.$

With these definitions we can prove the following existence criteria for non-
negative polynomials in a T-systems on $[a,b]$.

######  Theorem 4.3.28.

Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system on
$\mathcal{X}=[a,b]$, and $x_{1},\dots,x_{m}\in\mathcal{X}$. The following are
equivalent:

  1. (i)

The points $x_{1},\dots,x_{m}$ are zeros of a non-negative polynomial
$f\in\mathrm{lin}\,\mathcal{F}$.

  2. (ii)

$\displaystyle\sum_{i=1}^{m}\varepsilon(x_{i})\leq n$.

The proof is adapted from [21, p. 35, Thm. 1.2].

###### Proof.

“(i) $\Rightarrow$ (ii)” is 4.3.24 and we therefore only have to prove “(ii)
$\Rightarrow$ (i)”.

Case I: At first assume that $a<x_{1}<\dots<x_{m}<b$ and
$\sum_{i=0}^{m}\varepsilon(x_{i})=2m=n$. If $2m<n$ then add $k$ additional
points $x_{m+1},\dots,x_{m+k}$ such that $2m+2k=n$ and
$x_{m}<x_{m+1}<\dots<x_{m+k}<b$.

Select a sequence of points
$(x_{1}^{(j)},\dots,x_{m}^{(j)})\in\mathds{R}^{m}$, $j\in\mathds{N}$, such
that

$a<x_{1}<x_{1}^{(j)}<\dots<x_{m}<x_{m}^{(j)}<b$

for all $j\in\mathds{N}$ and $\lim_{j\to\infty}x_{i}^{(j)}=x_{i}$ for all
$i=1,\dots,m$. Set

$g_{j}(x):=c_{j}\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&\dots&f_{2m-1}&f_{2m}\\\
x&x_{1}&x_{1}^{(j)}&\dots&x_{m}&x_{m}^{(j)}\end{pmatrix}\quad\in(\mathrm{lin}\,\mathcal{F})^{e}$
(4)

for some $c_{j}>0$. Since $(\mathrm{lin}\,\mathcal{F})^{e}$ is compact we can
assume that $g_{j}$ converges to some
$g_{0}\in(\mathrm{lin}\,\mathcal{F})^{e}$. Then $g_{0}$ has
$x_{1},\dots,x_{m}$ as zeros with $\varepsilon(x_{i})=2$ and $g_{0}$ is non-
negative since $g_{j}>0$ on $[a,x_{1})$, $(x_{1}^{(j)},x_{2})$, …,
$(x_{m-1}^{(j)},x_{m})$, and $(x_{m}^{(j)},b]$ as well as $g_{j}<0$ on
$(x_{1},x_{1}^{(j)})$, $(x_{2},x_{2}^{(j)})$, …, $(x_{m},x_{m}^{(j)})$.

Case II: If $a=x_{1}<x_{2}<\dots<x_{m}<b$ with
$\sum_{i=1}^{m}\varepsilon(x_{i})=2m-1=n$ the only modification required in
case I is to replace (4) by

$g_{j}(x):=-c_{j}\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&f_{3}&\dots&f_{2m-2}&f_{2m-1}\\\
x&a&x_{2}&x_{2}^{(j)}&\dots&x_{m}&x_{m}^{(j)}\end{pmatrix}\quad\in(\mathrm{lin}\,\mathcal{F})^{e}$

with some normalizing factor $c_{j}>0$.

Case III: The procedure is similar if $x_{m}=b$ and
$\sum_{i=1}^{m}\varepsilon(x_{i})=n$. ∎

######  Remark 4.3.29.

4.3.28 appears in [21, p. 35, Thm. 1.2] in a stronger version. In [21, p. 35,
Thm. 1.2] Krein claims that the $x_{1},\dots,x_{m}$ are the only zeros of some
non-negative $f\in\mathrm{lin}\,\mathcal{F}$. This holds when $n=2m+2p$ for
some $p>0$. To see this add to $x_{1},\dots,x_{m}$ in (4) points
$x_{m+1},\dots,x_{m+p}\in\mathrm{int}\,\mathcal{X}\setminus\\{x_{1},\dots,x_{m}\\}$
and get $g_{0}$. Hence, $g_{0}\geq 0$ has exactly the zeros
$x_{1},\dots,x_{m+p}$. Then construct in a similar way $g_{0}^{\prime}$ with
the zeros $x_{1},\dots,x_{m},x^{\prime}_{m+1},\dots,x^{\prime}_{m+p}$ with
$x^{\prime}_{m+1},\dots,x^{\prime}_{m+p}\in\mathrm{int}\,\mathcal{X}\setminus\\{x_{1},\dots,x_{m+p}\\}$.
Hence, $g_{0}+g_{0}^{\prime}\geq 0$ has only the zeros $x_{1},\dots,x_{m}$.

A similar construction works for $n=2m+1$ with or without end points $a$ or
$b$.

However, Krein misses that for $n=2m$ and when one end point is contained in
$x_{1},\dots,x_{m}$ then it might happen that also the other end point must
appear. In [23, p. 28, Thm. 5.1] additional conditions are given which ensure
that $x_{1},\dots,x_{m}$ are the only zeros of some $f\geq 0$.

For example if also $\\{f_{i}\\}_{i=0}^{n-1}$ is a T-system then is can be
ensured that $x_{1},\dots,x_{m}$ are the only zeros of some non-negative
polynomial $f\in\mathrm{lin}\,\mathcal{F}$, see [23, p. 28, Thm. 5.1 (b-i)].
For our main example(s), the algebraic polynomials with gaps, this holds.

The same problem appears in [21, p. 36, Thm. 1.3]. A weaker but correct
version is given in 4.3.32 below. $\circ$

######  Remark 4.3.30.

Assume that in 4.3.28 we have additionally that $f_{0},\dots,f_{n}\in
C^{1}([a,b])$. Then in (4) we can set $x_{i}^{(j)}=x_{i}+j^{-1}$ for all
$i=0,\dots,m$ and $j\gg 1$. For $j\to\infty$ with $c_{j}=j^{m}$ we then get

$\displaystyle g_{0}(x)$
$\displaystyle=\lim_{j\to\infty}j^{m}\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&\dots&f_{2m-1}&f_{2m}\\\
x&x_{1}&x_{1}+j^{-1}&\dots&x_{m}&x_{m}+j^{-1}\end{pmatrix}$
$\displaystyle=\lim_{j\to\infty}j^{m}\cdot\det\begin{pmatrix}f_{0}(x)&\dots&f_{2m}(x)\\\
f_{0}(x_{1})&\dots&f_{2m}(x_{1})\\\
f_{0}(x_{1}+j^{-1})&\dots&f_{2m}(x_{1}+j^{-1})\\\ \vdots&&\vdots\\\
f_{0}(x_{m})&\dots&f_{2m}(x_{m})\\\
f_{0}(x_{m}+j^{-1})&\dots&f_{2m}(x_{m}+j^{-1})\\\ \end{pmatrix}$
$\displaystyle=\lim_{j\to\infty}\det\begin{pmatrix}f_{0}(x)&\dots&f_{2m}(x)\\\
f_{0}(x_{1})&\dots&f_{2m}(x_{1})\\\
\frac{f_{0}(x_{1}+j^{-1})-f_{0}(x_{1})}{j^{-1}}&\dots&\frac{f_{2m}(x_{1}+j^{-1})-f_{2m}(x_{1})}{j^{-1}}\\\
\vdots&&\vdots\\\ f_{0}(x_{m})&\dots&f_{2m}(x_{m})\\\
\frac{f_{0}(x_{m}+j^{-1})-f_{0}(x_{m})}{j^{-1}}&\dots&\frac{f_{2m}(x_{m}+j^{-1})-f_{2m}(x_{m})}{j^{-1}}\end{pmatrix}$
(5) $\displaystyle=\begin{pmatrix}f_{0}(x)&\dots&f_{2m}(x)\\\
f_{0}(x_{1})&\dots&f_{2m}(x_{1})\\\
f_{0}^{\prime}(x_{1})&\dots&f_{2m}^{\prime}(x_{1})\\\ \vdots&&\vdots\\\
f_{0}(x_{m})&\dots&f_{2m}(x_{m})\\\
f_{0}^{\prime}(x_{m})&\dots&f_{2m}^{\prime}(x_{m})\end{pmatrix},$

i.e., double zeros are included by including the values
$f_{i}^{\prime}(x_{j})$. Therefore, whenever we have $C^{1}$-functions in
$\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ and $x_{i}=x_{i+1}$ we define

$\begin{pmatrix}f_{0}&\dots&f_{i-1}&f_{i}&f_{i+1}&f_{i+2}&\dots&f_{n}\\\
x_{0}&\dots&x_{i-1}&(x_{i}&x_{i})&x_{i+2}&\dots&x_{n}\end{pmatrix}:=\begin{pmatrix}f_{0}(x_{0})&\dots&f_{n}(x_{0})\\\
\vdots&&\vdots\\\ f_{0}(x_{i-1})&\dots&f_{n}(x_{i-1})\\\
f_{0}(x_{i})&\dots&f_{n}(x_{i})\\\
f_{0}^{\prime}(x_{i})&\dots&f_{n}^{\prime}(x_{i})\\\
f_{0}(x_{i+2})&\dots&f_{n}(x_{i+2})\\\ \vdots&&\vdots\\\
f_{0}(x_{n})&\dots&f_{n}(x_{n})\end{pmatrix}$ (6)

and equivalently when $x_{j}=x_{j+1}$, $x_{k}=x_{k+1}$, … for additional
entries. We use the additional brackets “$($” and “$)$” to indicate that
$x_{i}$ is inserted in the $f_{0},\dots,f_{n}$ and then also into
$f_{0}^{\prime},\dots,f_{n}^{\prime}$ to distinguish (6) from 4.1.10 to avoid
confusion. Hence,

$\det\begin{pmatrix}f_{0}&\dots&f_{i-1}&f_{i}&f_{i+1}&f_{i+2}&\dots&f_{n}\\\
x_{0}&\dots&x_{i-1}&x_{i}&x_{i}&x_{i+2}&\dots&x_{n}\end{pmatrix}=0$

since in two rows $x_{i}$ is inserted into $f_{0},\dots,f_{n}$, while in

$\begin{pmatrix}f_{0}&\dots&f_{i-1}&f_{i}&f_{i+1}&f_{i+2}&\dots&f_{n}\\\
x_{0}&\dots&x_{i-1}&(x_{i}&x_{i})&x_{i+2}&\dots&x_{n}\end{pmatrix}$

indicates that $x_{i}$ is inserted in $f_{0},\dots,f_{n}$ and then also into
$f_{0}^{\prime},\dots,f_{n}^{\prime}$.

Extending this to zeros of order $k$ for $C^{k+1}$-functions is straight
forward and we leave it to the reader to write down the formulas and their
proofs. Similar to (6) we write for any $a\leq x_{0}\leq x_{1}\leq\dots\leq
x_{n}\leq b$ the matrix as

$\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}^{*}$

when $f_{0},\dots,f_{n}$ are sufficiently differentiable.

We often want to express polynomials $f\in\mathrm{lin}\,\mathcal{F}$ as
determinants (4) only by knowing their zeros $x_{1},\dots,x_{k}$. If arbitrary
multiplicities appear we only have $x_{1}\leq x_{2}\leq\dots\leq x_{n}$ where
we include zeros multiple times according to their multiplicities. Hence, for

$x_{0}=\dots=x_{i_{1}}<x_{i_{1}+1}=\dots=x_{i_{2}}<\dots<x_{i_{k}+1}=\dots=x_{n}$

we introduce a simpler notation to write down (6):

$\left(\begin{array}[]{c|cccc}f_{0}&f_{1}&f_{2}&\dots&f_{n}\\\
x&x_{1}&x_{2}&\dots&x_{n}\end{array}\right):=\begin{pmatrix}f_{0}&f_{1}\;\dots\;f_{i_{1}}&f_{i_{1}+1}\;\dots\;f_{i_{2}}&\dots&f_{i_{k}+1}\;\dots\;f_{i_{k}+1}\\\
x&(x_{1}\;\dots\;x_{i_{1}})&(x_{i_{1}+1}\;\dots\;x_{i_{2}})&\dots&(x_{i_{k}+1}\;\dots\;\;\;x_{n})\;\;\,\end{pmatrix}.$
(7)

Clearly $(\ref{eq:doublezeroDfn2})\in\mathrm{lin}\,\mathcal{F}$. For (7) to be
well-defined we need $\mathcal{F}\subset C^{m-1}$ where $m$ is the largest
multiplicity of any zero. However, the procedure (5) can lead to the zero
polynomial. We have to introduce ET-systems, see Section 4.4 and 4.4.34.

In 4.3.28 we did not need the condition $\mathcal{F}\subset C^{m}$ for some
$m\geq 1$. The limit $g_{0}$ of the $g_{j}$ in (4) does not need the unique
$f_{0}^{\prime},\dots,f_{n}^{\prime}$ and therefore the limit needs not to be
unique. $\circ$

######  Corollary 4.3.31.

4.3.28 also holds for intervals $\mathcal{X}\subseteq\mathds{R}$, i.e.,

$\mathcal{X}=(a,b),\ (a,b],\ [a,b),\ [a,b],\ (a,\infty),\ [a,\infty),\
(-\infty,b),\ (-\infty,b],\ \text{and}\ \mathds{R}\qquad\text{with}\ a<b.$ (8)

###### Proof.

We have that “(i) $\Rightarrow$ (ii)” follows from 4.3.25.

For “(ii) $\Rightarrow$ (i)” we apply 4.3.28 on
$[\min_{i}x_{i},\max_{i}x_{i}]$ and 4.3.25 assures that no additional zeros
appear in $\mathcal{X}$. ∎

We will give a sharper version of 4.3.24, see also 4.3.29.

######  Theorem 4.3.32.

Let $n\in\mathds{N}$ and $\mathcal{F}$ be a continuous T-system on
$\mathcal{X}=[a,b]$. Additionally, let $x_{1},\dots,x_{k}\in\mathcal{X}$ and
$y_{1},\dots,y_{l}\in\mathcal{X}$ be pairwise distinct points. The following
are equivalent:

  1. (i)

There exists a polynomial $f\in\mathrm{lin}\,\mathcal{F}$ such that

    1. (a)

$x_{1},\dots,x_{k}$ are the non-nodal zeros of $f$ and

    2. (b)

$y_{1},\dots,y_{l}$ are the nodal zeros of $f$.

  2. (ii)

$2k+l\leq n$.

###### Proof.

(i) $\Rightarrow$ (ii): That is 4.3.24.

(ii) $\Rightarrow$ (i): Adapt the proof and especially the $g_{j}$’s in (4) of
4.3.28 accordingly. Let $z_{1}<\dots<z_{k+l}$ be the $x_{i}$’s and $y_{i}$’s
together ordered by size. Then in $g_{j}$ treat every nodal $z_{i}$ like the
endpoint $a$ or $b$, i.e., include it only once in the determinant, and insert
for every non-nodal point $z_{i}$ the point $z_{i}$ and the sequence
$z_{i}^{(j)}\in(z_{i},z_{i+1})$ with $\lim_{j\to\infty}z_{i}^{(j)}=z_{i}$. ∎

######  Corollary 4.3.33.

4.3.32 also holds for sets $\mathcal{X}\subseteq\mathds{R}$ of the form

  1. (i)

$\mathcal{X}=(a,b)$, $[a,b)$, $(a,b]$ with $a<b$,

  2. (ii)

$\mathcal{X}=(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$,

  3. (iii)

$\mathcal{X}=\\{x_{1},\dots,x_{k}\\}\subseteq\mathds{R}$ with $k\geq n+1$ and
$x_{1}<\dots<x_{k}$, and

  4. (iv)

countable unions of (i) to (iii).

###### Proof.

In the adapted proof and the $g_{j}$’s in (4) of 4.3.28 we do not need to have
non-negativity, i.e., in the $g_{j}$’s sign changes at the $y_{i}$’s are
allowed (and even required). ∎

###  4.4 ET-Systems

######  Definition 4.4.34.

Let $n\in\mathds{N}$ and let $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}\subset
C^{n}([a,b])$ be a T-system of order $n$ on $[a,b]$ with $a<b$. $\mathcal{F}$
is called an _extended Tchebycheff system_ (_ET-system_) _of order $n$_ if any
polynomial $f\in\mathrm{lin}\,\mathcal{F}\setminus\\{0\\}$ has at most $n$
zeros counting algebraic multiplicities.

For notation of the matrices

$\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}^{*}$

for $a\leq x_{0}\leq x_{1}\leq\dots x_{n}\leq b$ see the previous 4.3.30.

######  Corollary 4.4.35 ([22] or e.g. [21, p. 37, P.1.1]).

Let $n\in\mathds{N}$ and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}\subset
C^{n}([a,b])$. The following are equivalent:

  1. (i)

$\mathcal{F}$ is an ET-system.

  2. (ii)

We have

$\det\begin{pmatrix}f_{0}&f_{1}&\dots&f_{n}\\\
x_{0}&x_{1}&\dots&x_{n}\end{pmatrix}^{*}\neq 0$

for every $a\leq x_{0}\leq x_{1}\leq\dots\leq x_{n}\leq b$.

###### Proof.

Follows immediately from 4.3.30. ∎

######  Example 4.4.36 (see e.g. [23, p. 19, Exm. 12]).

Let $n\in\mathds{N}$ and $g_{0},\dots,g_{n}\in C^{n}([a,b])$ such that
$g_{0},\dots,g_{n}>0$ on $[a,b]$ with $a<b$. Define

$\displaystyle f_{0}(x)$ $\displaystyle:=g_{0}(x)$ $\displaystyle f_{1}(x)$
$\displaystyle:=g_{0}(x)\cdot\int_{a}^{x}g_{1}(y_{1})\leavevmode\nobreak\
\mathrm{d}y_{1}$ $\displaystyle f_{2}(x)$
$\displaystyle:=g_{0}(x)\cdot\int_{a}^{x}g_{1}(y_{1})\cdot\int_{a}^{y_{1}}g_{2}(y_{2})\leavevmode\nobreak\
\mathrm{d}y_{2}\leavevmode\nobreak\ \mathrm{d}y_{1}$
$\displaystyle\;\;\;\vdots$ $\displaystyle f_{n}(x)$
$\displaystyle:=g_{0}(x)\cdot\int_{a}^{x}g_{1}(y_{1})\cdot\int_{a}^{y_{1}}g_{2}(y_{2})\leavevmode\nobreak\
{\dots}\int_{a}^{y_{n-1}}g_{n}(y_{n})\leavevmode\nobreak\
\mathrm{d}y_{n}\leavevmode\nobreak\ \dots\leavevmode\nobreak\
\mathrm{d}y_{2}\leavevmode\nobreak\ \mathrm{d}y_{1}.$

Then $\\{f_{i}\\}_{i=0}^{n}$ is an ET-system on $[a,b]$. $\circ$

######  Example 4.4.37.

Let $\mathcal{F}=\\{1,x,x^{3}\\}$ on $[0,b]$, $b>0$. Then $\mathcal{F}$ is a
T-system (Example 4.2.16(b)) but not an ET-system. To see this let
$x_{0}=x_{1}=x_{2}=0$, then

$\begin{pmatrix}f_{0}&f_{1}&f_{2}\\\
0&0&0\end{pmatrix}^{*}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix}.$

This shows that $\mathcal{F}$ is not an ET-system. $\circ$

In the previous example the position $x=0$ prevents the T-system to be a ET-
system. If $x=0$ is removed then it is an ET-system.

######  Example 4.4.38.

Let $\alpha_{0},\dots,\alpha_{n}\in\mathds{N}_{0}$ with
$\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$. Then
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$ on $[a,\infty)$ with $a>0$ is an
ET-system. For $n=2m$ and $a<x_{1}<x_{2}<\dots<x_{m}$ we often encounter a
specific polynomial structure and hence we write it down explicitly once:

$\displaystyle\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}$
$\displaystyle=\lim_{\varepsilon\to
0}\varepsilon^{-m}\cdot\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&x_{1}&x_{1}+\varepsilon&\dots&x_{m}&x_{m}+\varepsilon\end{pmatrix}$
$\displaystyle=\lim_{\varepsilon\to
0}\left[\prod_{i=1}^{m}(x_{i}-x)(x_{i}+\varepsilon-x)\right]\cdot\left[\prod_{1\leq
i<j\leq m}(x_{j}-x_{i})^{2}(x_{j}-x_{i}-\varepsilon)(x_{j}+\varepsilon-
x_{i})\right]$ (9) $\displaystyle\qquad\times
s_{\alpha}(x,x_{1},x_{1}+\varepsilon,\dots,x_{m},x_{m}+\varepsilon)$
$\displaystyle=\prod_{i=1}^{m}(x_{i}-x)^{2}\cdot\prod_{1\leq i<j\leq
m}(x_{j}-x_{i})^{4}\cdot s_{\alpha}(x,x_{1},x_{1},\dots,x_{m},x_{m})$

where $s_{\alpha}$ is the Schur polynomial of
$\alpha=(\alpha_{0},\dots,\alpha_{n})$ [26]. Hence,
$s_{\alpha}(x,x_{1},x_{1},\dots,x_{m},x_{m})$ is not divisible by some
$(x_{i}-x)$.

In fact, this is a special case of 4.4.36. With 4.4.36 we can even allow
$-\infty<\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}<\infty$ to be reals since
$a>0$. $\circ$

###### Proof.

Combine the induction

$f^{(m+1)}(x)=\lim_{h\to 0}\frac{f^{(m)}(x+h)-f^{(m)}(x)}{h}$

and

$\det\begin{pmatrix}x^{\alpha_{0}}&\dots&x^{\alpha_{n}}\\\
x_{0}&\dots&x_{n}\end{pmatrix}=\prod_{0\leq i<j\leq n}(x_{j}-x_{i})\cdot
s_{\alpha}(x_{0},\dots,x_{n})$

where $s_{\alpha}$ is the Schur polynomial of
$\alpha=(\alpha_{0},\dots,\alpha_{n})$. ∎

######  Example 4.4.39.

Let $n\in\mathds{N}$. Then the T-system $\mathcal{F}=\\{x^{i}\\}_{i=0}^{n}$ on
$\mathds{R}$ is an ET-system. $\circ$

##  5 Sparse Positivstellensätze and Nichtnegativstellensätze

In this section we present the Positivestellensatz for T-systems of Karlin
(5.1.41 and 5.1.42). We show their application to gain algebraic sparse
Positivestellensätze and Nichtnegativstellensätze. They will be used to solve
sparse moment problems.

###  5.1 Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$
for general T-Systems

For what follows we want to remind the reader of the index $\varepsilon(x)$ of
a point $x$, see 4.3.26.

######  Definition 5.1.40.

Let $\mathcal{Z}\subset[a,b]$. We say _$\mathcal{Z}$ has index $n$_ if
$\sum_{x\in\mathcal{Z}}\varepsilon(x)=n$. The same definition holds for sets
$\mathcal{X}$ as in 4.3.25.

Because of its importance and since it was first proved in full generality by
Karlin in [20] we call the following result Karlin’s Theorem.

######  Karlin’s Theorem 5.1.41 ([20] or e.g. [23, p. 66, Thm. 10.1]).

Let $n\in\mathds{N}$, $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a continuous
T-system of order $n$ on $[a,b]$ with $a<b$, and let $f\in C([a,b])$ with
$f>0$ on $[a,b]$. The following hold:

  1. (i)

There exists a unique polynomial $f_{*}\in\mathrm{lin}\,\mathcal{F}$ such that

    1. (a)

$f(x)\geq f_{*}(x)\geq 0$ for all $x\in[a,b]$,

    2. (b)

$f_{*}$ vanishes on a set with index $n$,

    3. (c)

the function $f-f_{*}$ vanishes at least once between each pair of adjacent
zeros of $f_{*}$,

    4. (d)

the function $f-f_{*}$ vanishes at least once between the largest zero of
$f_{*}$ and the end point $b$,

    5. (e)

$f_{*}(b)>0$.

  2. (ii)

There exists a unique polynomial $f^{*}\in\mathrm{lin}\,\mathcal{F}$ which
satisfies the conditions (a)–(d) of (i), and

    1. (e’)

$f^{*}(b)=0$.

###### Proof.

See e.g. [23, p. 68–71]. ∎

Note, in the previous result we do not need to have
$f\in\mathrm{lin}\,\mathcal{F}$. The function $f$ only needs to be continuous
and strictly positive on $[a,b]$.

An earlier version of 5.1.41 is a lemma by Markov [29], see also [43, p. 80].

For the same reason as for 5.1.41 we call the following immediate consequence
Karlin’s Corollary. It is the T-system Positivstellensatz by Karlin and will
be used to generate (algebraic) Positivestellensätze.

######  Karlin’s Corollary 5.1.42 ([20] or e.g. [23, p. 71, Cor. 10.1(a)]).

Let $n\in\mathds{N}$, $\mathcal{F}$ be a continuous T-system of order $n$ on
$[a,b]$ with $a<b$, and let $f\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on
$[a,b]$. Then there exists a unique representation

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ such that

  1. (i)

$f_{*},f^{*}\geq 0$ on $[a,b]$,

  2. (ii)

the zeros of $f_{*}$ and $f^{*}$ each are sets of index $n$,

  3. (iii)

the zeros of $f_{*}$ and $f^{*}$ strictly interlace, and

  4. (iv)

$f^{*}(b)=0$.

###### Proof.

Let $f_{*}$ be the unique $f_{*}$ from 5.1.41(i). Then
$f-f_{*}\in\mathrm{lin}\,\mathcal{F}$ is a polynomial and fulfills (a)–(d),
and (e’) of $f^{*}$ in 5.1.41. But since $f^{*}$ is unique we have
$f-f_{*}=f^{*}$. ∎

######  Corollary 5.1.43 ([20] or e.g. [23, p. 71, Cor. 10.1(b)]).

Let $n\in\mathds{N}$, $\\{f_{i}\\}_{i=0}^{n}$ and $\\{f_{i}\\}_{i=0}^{n+1}$ be
continuous T-systems of order $n$ and $n+1$ on $[a,b]$ with $a<b$. Then
$f_{n+1}-(f_{n+1})_{*}$ and $f_{n+1}-(f_{n+1})^{*}$ both vanish on sets of
index $n+1$.

######  Theorem 5.1.44 ([20] or e.g. [23, p. 72, Thm. 10.2]).

Let $n\in\mathds{N}$, $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a continuous
T-system of order $n$ on $[a,b]$ with $a<b$, and let $g_{1},g_{2}$ be two
continuous functions on $[a,b]$ such that there exists a
$g^{\prime}\in\mathrm{lin}\,\mathcal{F}$ with

$g_{1}(x)<g^{\prime}(x)<g_{2}(x)$

for all $x\in[a,b]$. The following hold:

  1. (i)

There exists a unique polynomial $f_{*}\in\mathrm{lin}\,\mathcal{F}$ such that

    1. (a)

$g_{1}(x)\leq f_{*}(x)\leq g_{2}(x)$ for all $x\in[a,b]$, and

    2. (b)

there exist $n+1$ points $x_{1}<\dots<x_{n+1}$ in $[a,b]$ such that

$f_{*}(x_{n+1-i})=\begin{cases}g_{1}(x_{n+1-i})&\text{for}\ i=1,3,5,\dots,\\\
g_{2}(x_{n+1-i})&\text{for}\ i=0,2,4,\dots.\end{cases}$

  2. (ii)

There exists a unique polynomial $f^{*}\in\mathrm{lin}\,\mathcal{F}$ such that

    1. (a’)

$g_{1}(x)\leq f^{*}(x)\leq g_{2}(x)$ for all $x\in[a,b]$, and

    2. (b’)

there exist $n+1$ points $y_{1}<\dots<y_{n+1}$ in $[a,b]$ such that

$f^{*}(y_{n+1-i})=\begin{cases}g_{2}(y_{n+1-i})&\text{for}\ i=1,3,5,\dots,\\\
g_{1}(y_{n+1-i})&\text{for}\ i=0,2,4,\dots.\end{cases}$

###### Proof.

See [23, p. 73]. ∎

In 5.1.41 and 5.1.42 we dealt with $f\in\mathrm{lin}\,\mathcal{F}$ with $f>0$,
i.e., they are the Positivstellensatz. The following result allows for $f\geq
0$ and is therefore together with 5.1.46 the T-system Nichtnegativstellensatz
of Karlin. We get from 5.1.46 sparse algebraic Nichtnegativstellensätze
(5.2.53).

######  Theorem 5.1.45 ([20] or e.g. [23, p. 74, Thm. 10.3]).

Let $n\in\mathds{N}$, $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a continuous ET-
system of order $n$ on $[a,b]$ with $a<b$, and let $f\in C^{n}([a,b])$ be such
that $f\geq 0$ on $[a,b]$ and $f$ has $r<n$ zeros (counting multiplicities).
The following hold:

  1. (i)

There exists a unique polynomial $f_{*}\in\mathrm{lin}\,\mathcal{F}$ such that

    1. (a)

$f(x)\geq f_{*}(x)\geq 0$ for all $x\in[a,b]$,

    2. (b)

$f_{*}$ has $n$ zeros counting multiplicities,

    3. (c)

if $x_{1}<\dots<x_{{n-r}}$ in $(a,b)$ are the zeros of $f_{*}$ which remain
after removing the $r$ zeros of $f$ then $f-f_{*}$ vanishes at least twice
more (counting multiplicities) in each open interval $(x_{i},x_{i+1})$,
$i=1,\dots,n-r-1$, and at least once more in each of the intervals $[a,x_{1})$
and $(x_{n-r},b]$,

    4. (d)

the zeros $x_{1},\dots,x_{n-r}$ of (c) are a set of index $n-r$, and

    5. (e)

$x_{n-r}<b$.

  2. (ii)

There exists a unique polynomial $f^{*}\in\mathrm{lin}\,\mathcal{F}$
satisfying the conditions (a) to (d) and (e’) $x_{n-r}=b$.

###### Proof.

See [23, p. 74–75]. ∎

######  Corollary 5.1.46 ([20] or e.g. [23, p. 76, Cor. 10.3]).

Let $n\in\mathds{N}$, $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be an ET-system of
order $n$ on $[a,b]$ with $a<b$, and let $f\in\mathrm{lin}\,\mathcal{F}$ be
such that $f\geq 0$ on $[a,b]$ and $f$ has $r<n$ zeros (counting
multiplicities). Then there exists a unique representation

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ such that

  1. (i)

$f_{*},f^{*}\geq 0$ on $[a,b]$,

  2. (ii)

$f_{*}$ and $f^{*}$ have $n$ zeros (counting multiplicity) which strictly
interlace if the zeros of $f$ are removed,

  3. (iii)

$f^{*}(b)=0$.

###  5.2 Sparse Positivstellensätze and Nichtnegativstellensätze on $[a,b]$
for Algebraic Polynomials

######  Theorem 5.2.47 (Sparse Algebraic Positivstellensatz on $[a,b]$ with
$0<a<b$).

Let $n\in\mathds{N}$, $\alpha_{0},\dots,\alpha_{n}\in\mathds{R}$ be real
numbers with $\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$, and let
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$. Then for any
$f=\sum_{i=0}^{n}a_{i}f_{i}\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on $[a,b]$
and $a_{n}>0$ there exists a unique decomposition

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ such that

  1. (i)

for $n=2m$ there exist points $x_{1},\dots,x_{m},y_{1},\dots,y_{m-1}\in[a,b]$
with

$a<x_{1}<y_{1}<\dots<x_{m}<b$

and constants $c_{*},c^{*}>0$ with

$f_{*}(x)=c_{*}\cdot\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\geq 0$

and

$f^{*}(x)=-c^{*}\cdot\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&a&(y_{1}&y_{1})&\dots&(y_{m-1}&y_{m-1})&b\end{pmatrix}\geq 0$

for all $x\in[a,b]$, or

  2. (ii)

for $n=2m+1$ there exist points $x_{1},\dots,x_{m},y_{1},\dots,y_{m}\in[a,b]$
with

$a<y_{1}<x_{1}<\dots<y_{m}<x_{m}<b$

and $c_{*},c^{*}>0$ with

$f_{*}(x)=-c_{*}\cdot\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&a&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\geq 0$

and

$f^{*}(x)=c^{*}\cdot\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(y_{1}&y_{1})&\dots&(y_{m}&y_{m})&b\end{pmatrix}\geq 0$

for all $x\in[a,b]$.

###### Proof.

By 4.4.38 we have that $\mathcal{F}$ on $[a,b]$ is an ET-system. Hence, 5.1.42
applies. We check both cases $n=2m$ and $n=2m+1$ separately.

$n=2m$: By 5.1.42 we have that the zero set $\mathcal{Z}(f^{*})$ of $f^{*}$
has index $2m$ and contains $b$ with index $1$, i.e., $a\in\mathcal{Z}(f^{*})$
and all other zeros have index $2$. Hence,
$\mathcal{Z}(f^{*})=\\{a=y_{0}<y_{1}<\dots<y_{m-1}<y_{m}=b\\}$. By 5.1.42 we
have that $\mathcal{Z}(f_{*})$ also has index $2m$ and the zeros of $f_{*}$
and $f^{*}$ interlace. Then the determinantal representations of $f_{*}$ and
$f^{*}$ follow from 4.3.30.

$n=2m+1$: By 5.1.42 we have that $b\in\mathcal{Z}(f^{*})$ and since the index
of $\mathcal{Z}(f^{*})$ is $2m+1$ we have that there are only double zeros
$y_{1},\dots,y_{m}\in(a,b)$ in $\mathcal{Z}(f^{*})$. Similar we find that
$a\in\mathcal{Z}(f_{*})$ since its index is odd and only double zeros
$x_{1},\dots,x_{m}\in(a,b)$ in $\mathcal{Z}(f_{*})$ remain. By 5.1.42(iii) the
zeros $x_{i}$ and $y_{i}$ strictly interlace and the determinantal
representation of $f_{*}$ and $f^{*}$ follow again from 4.3.30. ∎

Note, if $\alpha_{0},\dots,\alpha_{n}\in\mathds{N}_{0}$ then by 4.4.38
equation (9) the algebraic polynomials $f_{*}$ and $f^{*}$ can also be written
down with Schur polynomials.

5.2.47 does not to hold for $a=0$ and $\alpha_{0}>0$ or
$\alpha_{0},\dots,\alpha_{k}<0$. In case $\alpha_{0}>0$ the determinantal
representations of $f^{*}$ for $n=2m$ and $f_{*}$ for $n=2m+1$ are the zero
polynomial. In fact, in this case $\mathcal{F}$ is not even a T-system since
in 4.1.11 the determinant contains a zero column if $x_{0}=0$. We need to have
$\alpha_{0}=0$ ($x^{\alpha_{0}}=1$) to let $a=0$. For
$\alpha_{0},\dots,\alpha_{k}<0$ we have singularities at $x=0$ and hence no
T-system.

######  Corollary 5.2.48.

If $\alpha_{0}=0$ in 5.2.47 then 5.2.47 also holds with $a=0$.

###### Proof.

The determinantal representations of $f_{*}$ for $n=2m+1$ and $f^{*}$ for
$n=2m$ in 5.2.47 continuously depend on $a$. It is sufficient to show that
these representations are non-trivial (not the zero polynomial) for $a=0$. We
show this for $f_{*}$ in case (ii) $n=2m+1$. The other cases are equivalent.

For $\varepsilon>0$ small enough we set

$\displaystyle g_{\varepsilon}(x)$
$\displaystyle=-\varepsilon^{-m}\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&0&x_{1}&x_{1}+\varepsilon&\dots&x_{m}&x_{m}+\varepsilon\end{pmatrix}$
$\displaystyle=-\varepsilon^{-m}\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m+1}}\\\
1&0&0&\dots&0\\\
1&x_{1}^{\alpha_{1}}&x_{1}^{\alpha_{2}}&\dots&x_{1}^{\alpha_{2m+1}}\\\
\vdots&\vdots&\vdots&&\vdots\\\
1&(x_{m}+\varepsilon)^{\alpha_{1}}&(x_{m}+\varepsilon)^{\alpha_{2}}&\dots&(x_{m}+\varepsilon)^{\alpha_{2m+1}}\end{pmatrix}$
develop with respect to the second row
$\displaystyle=\varepsilon^{-m}\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}\\\
x_{1}^{\alpha_{1}}&x_{1}^{\alpha_{2}}&\dots&x_{1}^{\alpha_{2m-1}}\\\
\vdots&\vdots&&\vdots\\\
(x_{m}+\varepsilon)^{\alpha_{1}}&(x_{m}+\varepsilon)^{\alpha_{2}}&\dots&(x_{m}+\varepsilon)^{\alpha_{2m+1}}\end{pmatrix}$
$\displaystyle=\varepsilon^{-m}\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&x_{1}&x_{1}+\varepsilon&\dots&x_{m}&x_{m}+\varepsilon\end{pmatrix}.$

Then $x_{1},x_{1}+\varepsilon,\dots,x_{m},x_{m}+\varepsilon\in(0,b]$, i.e.,
$\\{x^{\alpha_{i}}\\}_{i=1}^{n}$ is an ET-system on $[a^{\prime},b]$ with
$0=a<a^{\prime}<x_{1}$, see 4.4.38. By 4.3.30 the representation is not the
zero polynomial which ends the proof. ∎

The 5.2.47 is a complete description of
$\mathrm{int}\,(\mathrm{lin}\,\mathcal{F})_{+}$. Since $\mathcal{F}$ is
continuous on the compact interval $[a,b]$ and $x^{\alpha_{0}}>0$ on $[a,b]$,
we have that the truncated moment cone is closed and hence
$(\mathrm{lin}\,\mathcal{F})_{+}$ and the moment cone are dual to each other.
With 5.2.47 we can now write down the conditions for the sparse truncated
Hausdorff moment problem on $[a,b]$ with $a>0$. We are not aware of a
reference for the following result.

######  Theorem 5.2.49 (Sparse Truncated Hausdorff Moment Problem on $[a,b]$
with $a>0$).

Let $n\in\mathds{N}$, $\alpha_{0},\dots,\alpha_{n}\in[0,\infty)$ with
$\alpha_{0}<\dots<\alpha_{n}$, and $a,b$ with $0<a<b$. Set
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$. Then the following are
equivalent:

  1. (i)

$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a truncated $[a,b]$-moment
functional.

  2. (ii)

$L(p)\geq 0$ holds for all

$\displaystyle p(x)$
$\displaystyle:=\begin{cases}\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\\\
-\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&a&(x_{1}&x_{1})&\dots&(x_{m-1}&x_{m-1})&b\end{pmatrix}\end{cases}$ if $n=2m$
or $\displaystyle p(x)$
$\displaystyle:=\begin{cases}-\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&a&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\\\
\det\begin{pmatrix}x^{\alpha_{0}}&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})&b\end{pmatrix}\end{cases}$ if $n=2m+1$

and all $x_{1},\dots,x_{m}$ with $a<x_{1}<\dots<x_{m}<b$.

###### Proof.

The implication (i) $\Rightarrow$ (ii) is clear since all given polynomials
$p$ are non-negative on $[a,b]$. It is therefore sufficient to prove (ii)
$\Rightarrow$ (i).

Since $a>0$ we have that $x^{\alpha_{0}}>0$ on $[a,b]$ and since $[a,b]$ is
compact we have that the moment cone $((\mathrm{lin}\,\mathcal{F})_{+})^{*}$
as the dual of the cone of non-negative (sparse) polynomials
$(\mathrm{lin}\,\mathcal{F})_{+}$ is a closed pointed cone.

To establish $L\in((\mathrm{lin}\,\mathcal{F})_{+})^{*}$ it is sufficient to
have $L(f)\geq 0$ for all $f\in(\mathrm{lin}\,\mathcal{F})_{+}$. Let
$f\in(\mathrm{lin}\,\mathcal{F})_{+}$. Then for all $\varepsilon>0$ we have
$f_{\varepsilon}:=f+\varepsilon\cdot x^{\alpha_{0}}>0$ on $[a,b]$, i.e., by
5.2.47 $f_{\varepsilon}$ is a conic combination of the polynomials $p$ in (ii)
and hence $L(f)+\varepsilon\cdot L(x^{\alpha_{0}})=L(f_{\varepsilon})\geq 0$
for all $\varepsilon>0$. Since $x^{\alpha_{0}}>0$ on $[a,b]$ we also have that
$x^{\alpha_{0}}$ is a conic combination of the polynomials $p$ in (ii) and
therefore $L(x^{\alpha_{0}})\geq 0$. Then $L(f)\geq 0$ follows from
$\varepsilon\to 0$ which proves (i). ∎

######  Corollary 5.2.50.

If $\alpha_{0}=0$ in 5.2.49 then 5.2.49 also holds with $a=0$, i.e., the
following are equivalent:

  1. (i)

$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a truncated $[0,b]$-moment
functional.

  2. (ii)

$L(p)\geq 0$ holds for all

$\displaystyle p(x)$
$\displaystyle:=\begin{cases}\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\\\
\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m-1}&x_{m-1})&b\end{pmatrix}\end{cases}$ if $n=2m$
or $\displaystyle p(x)$
$\displaystyle:=\begin{cases}\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\\\
\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})&b\end{pmatrix}\end{cases}$ if $n=2m+1$

and all $x_{1},\dots,x_{m}$ with $a<x_{1}<\dots<x_{m}<b$.

###### Proof.

Follows from 5.2.48. ∎

For the following we want to remind the reader of the Müntz–Szász Theorem [32,
47]. It states that for real exponents
$\alpha_{0}=0<\alpha_{1}<\alpha_{2}<\dots$ the vector space
$\mathrm{lin}\,\\{x^{\alpha_{i}}\\}_{i\in\mathds{N}_{0}}$ of finite linear
combinations is dense in $C([0,1])$ if and only if
$\sum_{i\in\mathds{N}}\frac{1}{\alpha_{i}}=\infty$.

We state the following only for the classical case of the interval $[0,1]$.
Other cases $[a,b]\subset[0,\infty)$ are equivalent. We are not aware of a
reference for the following result. Hausdorff required $\alpha_{i}\to\infty$.
The Müntz–Szász Theorem does not require $\alpha_{i}\to\infty$. The conditions
$\alpha_{0}=0$ and $\sum_{i\in\mathds{N}}\frac{1}{\alpha_{i}}=\infty$ already
appear in [14, eq. (17)].

######  Theorem 5.2.51 (Sparse Hausdorff Moment Problem).

Let $\\{\alpha_{i}\\}_{i\in\mathds{N}_{0}}\subset[0,\infty)$ with
$0=\alpha_{0}<\alpha_{1}<\dots$ and
$\sum_{i\in\mathds{N}}\frac{1}{\alpha_{i}}=\infty$. Let
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i\in\mathds{N}_{0}}$. The following are
equivalent:

  1. (i)

$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a $[0,1]$-moment functional.

  2. (ii)

$L(p)\geq 0$ holds for all $p\in(\mathrm{lin}\,\mathcal{F})_{+}$.

  3. (iii)

$L(p)\geq 0$ holds for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.

  4. (iv)

$L(p)\geq 0$ holds for all

$p(x)=\begin{cases}\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix},\\\
\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m-1}&x_{m-1})&1\end{pmatrix},\\\
\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix},\ \text{and}\\\
\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})&1\end{pmatrix}\end{cases}$

for all $m\in\mathds{N}$ and all $0<x_{1}<x_{2}<\dots<x_{m}<1$.

###### Proof.

The implications “(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)” are clear
and “(iii) $\Leftrightarrow$ (iv)” follows from 5.2.47. It is therefore
sufficient to show “(ii) $\Rightarrow$ (i)”.

Let $f\in C([0,1])$ with $f>0$. Since $\mathrm{lin}\,\mathcal{F}$ is dense in
$C([0,1])$ by the Müntz–Szász Theorem there are sequences
$\\{g_{i}\\}_{i\in\mathds{N}_{0}}$ and $\\{h_{i}\\}_{i\in\mathds{N}_{0}}$ with
$0<g_{i}<f<h_{i}$ and $\|g_{i}-h_{i}\|_{\infty}\to 0$ as $i\to\infty$. Hence,
$L(f)\geq 0$. Since $f\in C([0,1])$ with $f>0$ was arbitrary we have that
$L(f)\geq 0$ for all $f\in C([0,1])$ with $f\geq 0$. Then by the
Riesz–Markov–Kakutani Representation Theorem we have that $L$ has a unique
representing measure. ∎

The previous proof can be simplified by using Choquet’s theory of adapted
spaces, see [7] or for a more modern formulation [33] or [41, Ch. 1]. With
that we can even remove the use of the Müntz–Szász Theorem and therefore the
condition $\sum_{i\in\mathds{N}}\frac{1}{\alpha_{i}}=\infty$. Additionally, we
can allow for negative exponents. We will use this approach below and also in
all other proofs from here on. The following theorem has to our knowledge not
been presented before.

######  Theorem 5.2.52 (General Sparse Hausdorff Moment Problem on $[a,b]$
with $0\leq a<b$).

Let $I\subset\mathds{N}_{0}$ be an index set (finite or infinite), let
$\\{\alpha_{i}\\}_{i\in I}$ be such that $\alpha_{i}\neq\alpha_{j}$ for all
$i\neq j$ and

  1. (a)

if $a=0$ then $\\{\alpha_{i}\\}_{i\in I}\subset[0,\infty)$ with $\alpha_{i}=0$
for an $i\in I$, or

  2. (b)

if $a>0$ then $\\{\alpha_{i}\\}_{i\in I}\subset\mathds{R}$.

Let $\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i\in I}$. Then the following are
equivalent:

  1. (i)

$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a Hausdorff moment functional.

  2. (ii)

$L(p)\geq 0$ holds for all $p\in(\mathrm{lin}\,\mathcal{F})_{+}$.

  3. (iii)

$L(p)\geq 0$ holds for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.

  4. (iv)

$L(p)\geq 0$ holds for all

$p(x)=\begin{cases}\det\begin{pmatrix}x^{\alpha_{i_{0}}}&x^{\alpha_{i_{1}}}&x^{\alpha_{i_{2}}}&\dots&x^{\alpha_{i_{2m-1}}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix},&\text{if $|I|=2m$ or
$\infty$,}\\\
\det\begin{pmatrix}x^{\alpha_{i_{1}}}&x^{\alpha_{i_{2}}}&x^{\alpha_{i_{3}}}&\dots&x^{\alpha_{i_{2m-2}}}&x^{\alpha_{i_{2m-1}}}&x^{\alpha_{i_{2m}}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m-1}&x_{m-1})&b\end{pmatrix},&\text{if $|I|=2m$ or
$\infty$,}\\\
\det\begin{pmatrix}x^{\alpha_{i_{1}}}&x^{\alpha_{i_{2}}}&x^{\alpha_{i_{3}}}&\dots&x^{\alpha_{i_{2m}}}&x^{\alpha_{i_{2m+1}}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix},\ &\text{if $|I|=2m+1$ or
$\infty$, and}\\\
\det\begin{pmatrix}x^{\alpha_{i_{0}}}&x^{\alpha_{i_{1}}}&x^{\alpha_{i_{2}}}&\dots&x^{\alpha_{i_{2m-1}}}&x^{\alpha_{i_{2m}}}&x^{\alpha_{i_{2m+1}}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})&b\end{pmatrix},&\text{if $|I|=2m+1$ or
$\infty$,}\end{cases}$

for all $m\in\mathds{N}$ if $|I|=\infty$, all $0<x_{1}<x_{2}<\dots<x_{m}<b$,
and all $\alpha_{i_{0}}<\alpha_{i_{1}}<\dots<\alpha_{i_{m}}$ with
$\alpha_{i_{0}}=0$ if $a=0$.

If additionally $\sum_{i:\alpha_{i}\neq 0}\frac{1}{|\alpha_{i}|}=\infty$ then
$L$ is determinate.

###### Proof.

The case $|I|<\infty$ is 5.2.49. We therefore prove the case $|I|=\infty$. The
choice $\alpha_{i_{0}}<\alpha_{i_{1}}<\dots<\alpha_{i_{m}}$ with
$\alpha_{i_{0}}=0$ if $a=0$ makes $\\{x^{\alpha_{i_{j}}}\\}_{j=0}^{m}$ a
T-system. The implications “(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)”
are clear and “(iii) $\Leftrightarrow$ (iv)” is 5.2.47. It is therefore
sufficient to show “(ii) $\Rightarrow$ (i)”.

The space $\mathrm{lin}\,\mathcal{F}$ is an adapted space and the assertion
follows therefore from [41, Thm. 1.8].

For the determinacy of $L$ split $\\{\alpha_{i}\\}_{i\in I}$ into positive and
negative exponents. If $\sum_{i:\alpha_{i}\neq
0}\frac{1}{|\alpha_{i}|}=\infty$ then the corresponding sum over at least one
group is infinite. If the sum over the positive exponents is infinite apply
the Müntz–Szász Theorem. If the sum over the negative exponents is infinite
apply the Müntz–Szász Theorem to $\\{(x^{-1})^{-\alpha_{i}}\\}_{i\in
I:\alpha_{i}<0}$ since $a>0$. ∎

Note, since $[a,b]$ is compact the fact that $\\{x^{\alpha_{i}}\\}_{i\in I}$
is an adapted space is trivial.

In the previous results we only needed the description of all strictly
positive polynomials. The non-negative polynomials are described in the
following result. Again, we are not aware of a reference.

######  Theorem 5.2.53 (Sparse Algebraic Nichtnegativstellensatz on $[a,b]$
with $0<a<b$).

Let $n\in\mathds{N}$, $\alpha_{0},\dots,\alpha_{n}\in\mathds{R}$ be real
numbers with $\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$, and let
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$. Let
$f\in\mathrm{lin}\,\mathcal{F}$ with $f\geq 0$ on $[a,b]$. Then there exist
points $x_{1},\dots,x_{n},y_{1},\dots,y_{n}\in[a,b]$ (not necessarily
distinct) with $y_{n}=b$ which include the zeros of $f$ with multiplicities
and there exist constants $c_{*},c^{*}\in\mathds{R}$ such that

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$, $f_{*},f^{*}\geq 0$ on
$[a,b]$, and the polynomials $f_{*}$ and $f^{*}$ are given by

$f_{*}(x)=c_{*}\cdot\det\left(\begin{array}[]{c|ccc}f_{0}&f_{1}&\dots&f_{n}\\\
x&x_{1}&\dots&x_{n}\end{array}\right)\qquad\text{and}\qquad
f^{*}(x)=c_{*}\cdot\det\left(\begin{array}[]{c|ccc}f_{0}&f_{1}&\dots&f_{n}\\\
x&y_{1}&\dots&y_{n}\end{array}\right)$

for all $x\in[a,b]$.

Removing the zeros of $f$ from $x_{1},\dots,x_{n},y_{1},\dots,y_{n}$ we can
assume that the remaining $x_{i}$ and $y_{i}$ are disjoint and when grouped by
size the groups strictly interlace:

$a\ \leq\ x_{i_{1}}=\dots=x_{i_{k}}\ <\ y_{j_{1}}=\dots=y_{j_{l}}\ <\ \dots\
<x_{i_{p}}=\dots=x_{i_{q}}\ <\ y_{j_{r}}=\dots=y_{j_{s}}=b.$

Each such group in $(a,b)$ has an even number of members.

###### Proof.

By 4.4.38 we have that $\mathcal{F}$ on $[a,b]$ is an ET-system. We then apply
5.1.46 similar to the proof of 5.2.47. ∎

The signs of $c_{*}$ and $c^{*}$ are determined by $x_{1}$ and $y_{1}$ and
their multiplicity. If $x_{1}=\dots=x_{k}<x_{k+1}\leq\dots\leq x_{n}$ then
$\mathrm{sgn}\,c_{*}=(-1)^{k}$. The same holds for $c^{*}$ from the $y_{i}$.

######  Corollary 5.2.54.

If $\alpha_{0}=0$ in 5.2.53 then 5.2.53 also holds for $\alpha_{0}=0$.

######  Example 5.2.55.

Let $\alpha\in(0,\infty)$ and let $\mathcal{F}=\\{1,x^{\alpha}\\}$ on $[0,1]$.
Then we have $1=1_{*}+1^{*}$ with $1_{*}=x^{\alpha}$ and $1^{*}=1-x^{\alpha}$.
$\circ$

###  5.3 Sparse Positivstellensätze and Nichtnegativstellensätze on
$[0,\infty)$

In Section 5.1 we have seen the general Positivstellen- and
Nichtnegativitätsstellensätze for T-systems and then applied these to the
algebraic cases on $[a,b]$. We now show how the results from Section 5.1 on
$[a,b]$ can be transferred to $[0,\infty)$.

######  Theorem 5.3.56 ([20] or e.g. [23, p. 168, Thm. 8.1]).

Let $n\in\mathds{N}$ and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{n}$ be a continuous
T-system of order $n$ on $[0,\infty)$ such that

  1. (a)

there exists a $C>0$ such that $f_{n}(x)>0$ for all $x\geq C$,

  2. (b)

$\displaystyle\lim_{x\to\infty}\frac{f_{i}(x)}{f_{n}(x)}=0$ for all
$i=0,\dots,n-1$, and

  3. (c)

$\\{f_{i}\\}_{i=0}^{n-1}$ is a continuous T-system on $[0,\infty)$.

Then for any $f=\sum_{i=0}^{n}a_{i}f_{i}\in\mathrm{lin}\,\mathcal{F}$ with
$f>0$ on $[0,\infty)$ and $a_{n}>0$ there exists a unique representation

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ with $f_{*},f^{*}\geq 0$ on
$[0,\infty)$ such that the following hold:

  1. (i)

If $n=2m$ the polynomials $f_{*}$ and $f^{*}$ each possess $m$ distinct zeros
$\\{x_{i}\\}_{i=1}^{m}$ and $\\{y_{i}\\}_{i=0}^{m-1}$ satisfying

$0=y_{0}<x_{1}<y_{1}<\dots<y_{m-1}<x_{m}<\infty.$

All zeros except $y_{0}$ are double zeros.

  2. (ii)

If $n=2m+1$ the polynomials $f_{*}$ and $f^{*}$ each possess the zeros
$\\{x_{i}\\}_{i=1}^{m+1}$ and $\\{y_{i}\\}_{i=1}^{m}$ satisfying

$0=x_{1}<y_{1}<x_{2}<\dots<y_{m}<x_{m+1}<\infty.$

All zeros except $x_{1}$ are double zeros.

  3. (iii)

The coefficient of $f_{n}$ in $f_{*}$ is equal to $a_{n}$.

###### Proof.

By (a) there exists a function $w\in C([0,\infty))$ such that $w>0$ on
$[0,\infty)$ and $\lim_{x\to\infty}\frac{f_{n}(x)}{w(x)}=1$. By (b) we define

$v_{i}(x):=\begin{cases}\frac{f_{i}(x)}{w(x)}&\text{if}\ x\in[0,\infty),\\\
\delta_{i,n}&\text{if}\ x=\infty\end{cases}$

for all $i=0,1,\dots,n$. Then by (c) $\\{v_{i}\\}_{i=0}^{n}$ is a T-system on
$[0,\infty]$ by 4.2.20. With $t(x):=\tan(\pi x/2)$ we define
$g_{i}(x):=v_{i}\circ t$ for all $i=0,1,\dots,n$. Hence,
$\mathcal{G}=\\{g_{i}\\}_{i=0}^{n}$ is a T-system on $[0,1]$. We now apply
5.1.42 to $\mathcal{G}$. Set $g:=(\frac{f}{w})\circ t$.

(i): Let $n=2m$. Then by 5.1.42 there exits points

$0=y_{0}<x_{1}<y_{1}<\dots<x_{m}<y_{m}=1$

and unique functions $g_{*}$ and $g^{*}$ such that $g=g_{*}+g^{*}$,
$g_{*},g^{*}\geq 0$ on $[0,1]$, $x_{1},\dots,x_{m}$ are the zeros of $g_{*}$,
and $y_{0},\dots,y_{m}$ are the zeros of $g^{*}$. Then $f_{*}:=(g_{*}\circ
t^{-1})\cdot w$ and $f^{*}:=(g^{*}\circ t^{-1})\cdot w$ are the unique
components in the decomposition $f=f_{*}+f^{*}$.

(ii): Similar to (i).

(iii): From (i) (and (ii) in a similar way) we have $g_{i}(1)=0$ for
$i=0,\dots,n-1$ and $g_{n}(1)=1$. Hence, we get with $g^{*}(y_{m}=1)=0$ that
$g_{n}$ is not contained in $g^{*}$, i.e., $g_{*}$ has the only $g_{n}$
contribution because $\mathcal{G}$ is linearly independent. This is inherited
by $f_{*}$ and $f^{*}$ which proves (iii). ∎

If $\mathcal{F}$ is an ET-system then the $f_{*}$ and $f^{*}$ can be written
down explicitly.

######  Corollary 5.3.57.

If in 5.3.56 we have additionally that $\mathcal{F}$ is an ET-system on
$[0,\infty)$ then the unique $f_{*}$ and $f^{*}$ are given

  1. (i)

for $n=2m$ by

$f_{*}(x)=a_{2m}\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&\dots&f_{2m-1}&f_{2m}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}$

and

$f^{*}(x)=-c\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&f_{3}&\dots&f_{2m-2}&f_{2m-1}\\\
x&y_{0}&(y_{1}&y_{1})&\dots&(y_{m-1}&y_{m-1})\end{pmatrix},$

  2. (ii)

and for $n=2m+1$ by

$f_{*}(x)=-a_{2m+1}\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&f_{3}&\dots&f_{2m}&f_{2m+1}\\\
x&x_{1}&(x_{2}&x_{2})&\dots&(x_{m+1}&x_{m+1})\end{pmatrix}$

and

$f^{*}(x)=c\cdot\det\begin{pmatrix}f_{0}&f_{1}&f_{2}&\dots&f_{2m-1}&f_{2m}\\\
x&(y_{1}&y_{1})&\dots&(y_{m}&y_{m})\end{pmatrix}$

for some $c>0$.

###### Proof.

Combine 5.3.56 with 4.3.30. ∎

If we now plug 4.2.16 into 5.3.56 we get the following.

######  Theorem 5.3.58 (Sparse Algebraic Positivstellensatz on $[0,\infty)$).

Let $n\in\mathds{N}$, $\alpha_{0},\dots,\alpha_{n}\in[0,\infty)$ be real
numbers with $\alpha_{0}=0<\alpha_{1}<\dots<\alpha_{n}$, and let
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i=0}^{n}$ on $[0,\infty)$. Then for any
$f=\sum_{i=0}^{n}a_{i}f_{i}\in\mathrm{lin}\,\mathcal{F}$ with $f>0$ on
$[0,\infty)$ and $a_{n}>0$ there exists a unique decomposition

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ and $f_{*},f^{*}\geq 0$ on
$[0,\infty)$ such that the following hold:

  1. (i)

If $n=2m$ then the polynomials $f_{*}$ and $f^{*}$ each possess $m$ distinct
zeros $\\{x_{i}\\}_{i=1}^{m}$ and $\\{y_{i}\\}_{i=0}^{m-1}$ satisfying

$0=y_{0}<x_{1}<y_{1}<\dots<y_{m-1}<x_{m}<\infty.$

The polynomials $f_{*}$ and $f^{*}$ are given by

$f_{*}(x)=a_{2m}\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}$

and

$\displaystyle f^{*}(x)$
$\displaystyle=c\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}\\\
x&(y_{1}&y_{1})&\dots&(y_{m-1}&y_{m-1})\end{pmatrix}$

for some $c>0$.

  2. (ii)

If $n=2m+1$ then $f_{*}$ and $f^{*}$ have zeros $\\{x_{i}\\}_{i=1}^{m+1}$ and
$\\{y_{i}\\}_{i=1}^{m}$ respectively which satisfy

$0=x_{1}<y_{1}<x_{2}<\dots<y_{m}<x_{m+1}<\infty.$

The polynomials $f_{*}$ and $f^{*}$ are given by

$\displaystyle f_{*}(x)$
$\displaystyle=a_{2m+1}\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{2}&x_{2})&\dots&(x_{m+1}&x_{m+1})\end{pmatrix}$

and

$f^{*}(x)=c\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(y_{1}&y_{1})&\dots&(y_{m}&y_{m})\end{pmatrix}$

for some $c>0$.

###### Proof.

We have that $\mathcal{F}$ clearly fulfill condition (a) and (b) of 5.3.56 and
by 4.2.16 we known that $\mathcal{F}$ on $[0,\infty)$ is also a T-system,
i.e., (c) in 5.3.56 is fulfilled. We can therefore apply 5.3.56.

(i) $n=2m$: By 5.3.56(i) the unique $f_{*}$ and $f^{*}$ each possess $m$
distinct zeros $\\{x_{i}\\}_{i=1}^{m}$ and $\\{y_{i}\\}_{i=0}^{m-1}$ with
$0\leq y_{0}<x_{1}<\dots<y_{m-1}<x_{m}<\infty$. Since
$x_{1},\dots,x_{m}\in(0,\infty)$ and $\mathcal{F}$ on $[x_{1}/2,\infty)$ is an
ET-system we immediately get the determinantal representation of $f_{*}$ by
5.3.57 (combine 5.3.56 with 4.3.30). For $f^{*}$ we have $y_{0}=0$ and by
4.4.37 this is no ET-system. Hence, we prove the representation of $f^{*}$ by
hand.

Let $\varepsilon>0$ be such that
$0=y_{0}<y_{1}<y_{1}+\varepsilon<\dots<y_{m-1}<y_{m-1}+\varepsilon$ holds.
Then

$\displaystyle g_{\varepsilon}(x)$
$\displaystyle=-\varepsilon^{-m+1}\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}\\\
x&0&y_{1}&y_{1}+\varepsilon&\dots&y_{m-1}&y_{m-1}+\varepsilon\end{pmatrix}$
$\displaystyle=-\varepsilon^{-m+1}\cdot\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}\\\
1&0&0&\dots&0\\\
1&y_{1}^{\alpha_{1}}&y_{1}^{\alpha_{2}}&\dots&y_{1}^{\alpha_{2m-1}}\\\
\vdots&\vdots&\vdots&&\vdots\\\
1&(y_{m-1}+\varepsilon)^{\alpha_{1}}&(y_{m-1}+\varepsilon)^{\alpha_{2}}&\dots&(y_{m-1}+\varepsilon)^{\alpha_{2m-1}}\end{pmatrix}$
expand by the second row
$\displaystyle=\varepsilon^{-m+1}\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}\\\
y_{1}^{\alpha_{1}}&y_{1}^{\alpha_{2}}&\dots&y_{1}^{\alpha_{2m-1}}\\\
\vdots&\vdots&&\vdots\\\
(y_{m-1}+\varepsilon)^{\alpha_{1}}&(y_{m-1}+\varepsilon)^{\alpha_{2}}&\dots&(y_{m-1}+\varepsilon)^{\alpha_{2m-1}}\end{pmatrix}$
$\displaystyle=\varepsilon^{-m+1}\cdot\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}\\\
x&y_{1}&y_{1}+\varepsilon&\dots&y_{m-1}&y_{m-1}+\varepsilon\end{pmatrix}$

is non-negative on $[0,y_{1}]$ and every $[y_{i}+\varepsilon,y_{i+1}]$. Now
$y_{0}=0$ is removed and all $y_{i},y_{i}+\varepsilon>0$. Hence, we can work
on $[y_{1}/2,\infty)$ where $\\{x^{\alpha_{i}}\\}_{i=1}^{2m}$ is an ET-system
and we can go to the limit $\varepsilon\to 0$ as in 4.3.30. Then 5.3.57 proves
the representation of $f^{*}$.

(ii) $n=2m+1$: Similar to the case (i) with $n=2m$. ∎

The previous result was reproved in [2]. Additionally, since the authors of
[2] were not aware of [20, 23] their statement is much weaker and the proof is
unnecessary long and complicated. In [2] several other results are reproved
which already appeared in [23].

It is left to the reader to use 5.1.46 to gain the corresponding sparse
Nichtnegativestellensatz on $[0,\infty)$ for general T-systems and for
$\\{x^{\alpha_{i}}\\}_{i=0}^{n}$ with
$0=\alpha_{0}<\alpha_{1}<\dots<\alpha_{n}$ real numbers. The proofs follow the
same line of thoughts as the proof of 5.2.53. If all
$\alpha_{i}\in\mathds{N}_{0}$ then we can express the $f_{*}$ and $f^{*}$ in
5.3.58 also with Schur polynomials, see (9) in 4.4.38.

We have seen that Boas already investigated the sparse Stieltjes moment
problem [4]. However, since Boas did not have access to 5.3.56 by Karlin and
therefore 5.3.58 the description was complicated and incomplete. We get the
following complete and simple description. To our knowledge this result did
not appear somewhere else.

######  Theorem 5.3.59 (Sparse Stieltjes Moment Problem).

Let $\\{\alpha_{i}\\}_{i\in\mathds{N}_{0}}\subset[0,\infty)$ such that
$\alpha_{0}=0<\alpha_{1}<\alpha_{2}<\dots$ and let
$\mathcal{F}=\\{x^{\alpha_{i}}\\}_{i\in\mathds{N}_{0}}$. Then the following
are equivalent:

  1. (i)

$L:\mathrm{lin}\,\mathcal{F}\to\mathds{R}$ is a $[0,\infty)$-moment
functional.

  2. (ii)

$L(p)\geq 0$ for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p\geq 0$.

  3. (iii)

$L(p)\geq 0$ for all $p\in\mathrm{lin}\,\mathcal{F}$ with $p>0$.

  4. (iv)

$L(p)\geq 0$ for all

$p(x)=\begin{cases}\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix},\\\
\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m-2}}&x^{\alpha_{2m-1}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m-1}&x_{m-1})\end{pmatrix},\\\
\det\begin{pmatrix}x^{\alpha_{1}}&x^{\alpha_{2}}&x^{\alpha_{3}}&\dots&x^{\alpha_{2m}}&x^{\alpha_{2m+1}}\\\
x&(x_{2}&x_{2})&\dots&(x_{m+1}&x_{m+1})\end{pmatrix},\ \text{and}\\\
\det\begin{pmatrix}1&x^{\alpha_{1}}&x^{\alpha_{2}}&\dots&x^{\alpha_{2m-1}}&x^{\alpha_{2m}}\\\
x&(x_{1}&x_{1})&\dots&(x_{m}&x_{m})\end{pmatrix}\end{cases}$

for all $m\in\mathds{N}_{0}$ and $0<x_{1}<\dots<x_{m}$.

###### Proof.

The implications “(i) $\Rightarrow$ (ii) $\Leftrightarrow$ (iii)” are clear
and “(iii) $\Leftrightarrow$ (iv)” is 5.3.58. It is therefore sufficient to
prove “(ii) $\Leftarrow$ (i)”.

We have
$\mathrm{lin}\,\mathcal{F}=(\mathrm{lin}\,\mathcal{F})_{+}-(\mathrm{lin}\,\mathcal{F})_{+}$,
we have $1=x^{\alpha_{0}}\in\mathrm{lin}\,\mathcal{F}$, and for any
$g=\sum_{i=0}^{m}a_{i}\cdot x^{\alpha_{i}}\in(\mathrm{lin}\,\mathcal{F})_{+}$
we have $\lim_{x\to\infty}\frac{g(x)}{x^{\alpha_{m+1}}}=0$, i.e., there exists
an $f\in(\mathrm{lin}\,\mathcal{F})_{+}$ which dominates $g$. Hence,
$\mathrm{lin}\,\mathcal{F}$ is an adapted space and the assertion follows from
[41, Thm. 1.8]. ∎

Note, in the previous result we did needed
$0=\alpha_{0}<\alpha_{1}<\alpha_{2}<\dots$ but we did not need
$\alpha_{i}\to\infty$. Hence, 5.3.59 also includes the case
$\sup_{i\in\mathds{N}_{0}}\alpha_{i}<\infty$.

###  5.4 Sparse Positivstellensätze and Nichtnegativstellensätze on
$\mathds{R}$

######  Theorem 5.4.60 ([20] or e.g. [23, p. 198, Thm. 8.1]).

Let $m\in\mathds{N}_{0}$ and $\mathcal{F}=\\{f_{i}\\}_{i=0}^{2m}$ be a
continuous T-system of order $2m$ on $\mathds{R}$ such that

  1. (a)

there exists a $C>0$ such that $f_{2m}(x)>0$ for all
$x\in(-\infty,-C]\cup[C,\infty)$,

  2. (b)

$\displaystyle\lim_{|x|\to\infty}\frac{f_{i}(x)}{f_{2m}(x)}=0$ for all
$i=0,\dots,2m-1$, and

  3. (c)

$\\{f_{i}\\}_{i=0}^{2m-1}$ is a continuous T-system of order $2m-1$ on
$\mathds{R}$.

Let $f=\sum_{i=0}^{2m}a_{i}f_{i}$ be such that $f>0$ on $\mathds{R}$ and
$a_{2m}>0$. Then there exists a unique representation

$f=f_{*}+f^{*}$

with $f_{*},f^{*}\in\mathrm{lin}\,\mathcal{F}$ and $f_{*},f^{*}\geq 0$ on
$\mathds{R}$ such that

  1. (i)

the coefficient of $f_{2m}$ in $f_{*}$ is $a_{2m}$, and

  2. (ii)

$f_{*}$ and $f^{*}$ are non-negative polynomials having zeros
$\\{x_{i}\\}_{i=1}^{m}$ and $\\{y_{i}\\}_{i=1}^{m-1}$ with

$-\infty<x_{1}<y_{1}<x_{2}<\dots<y_{m-1}<x_{m}<\infty.$

###### Proof.

Adapt the proof of 5.3.56 such that both interval ends of $[a,b]$ are mapped
to $-\infty$ and $+\infty$, respectively. ∎

We have already seen how from 5.1.41 and 5.1.42 we gained 5.2.47 (sparse
algebraic Positivstellensatz on $[a,b]$), 5.2.53 (sparse algebraic
Nichtnegativstellensatz on $[a,b]$), and Theorems 5.2.51 and 5.2.52 (sparse
Hausdorff moment problems). We have seen how from 5.1.41 and 5.1.42 we gained
5.3.56 (sparse Positivstellensatz for T-systems on $[0,\infty)$), 5.3.58
(sparse algebraic Positivstellensatz on $[0,\infty)$), and 5.3.59 (sparse
Stieltjes moment problem). We will therefore not repeat this procedure for the
case $K=\mathds{R}$ from 5.4.60 but summarize the procedure in the following
“cooking receipt”.

######  Remark 5.4.61 (A General Cooking Receipt).

We have the following general _cooking receipt_ for generating sparse
Positivstellensätze, Nichtnegativstellensätze, and to generate and solve
sparse moment problems:

  1. (A)

Use 5.1.41 or 5.1.42, or extend these to sets $K=[a,b]\cup[c,d],\dots$ (for
extensions see e.g. [23] and later literature on T-systems we did not discuss
here).

  2. (B)

Prove that your family $\mathcal{F}=\\{f_{i}\\}_{i\in I}$ is a T-system (or
even an ET-system).

  3. (C)

Plug $\mathcal{F}$ into (A) to get the sparse Positivstellensatz or sparse
Nichtnegativestellensatz on $K$.

  4. (D)

Show that $\mathrm{lin}\,\mathcal{F}$ is an adapted space.

  5. (E)

Combine (C) and (D) to a sparse moment problem (use [41, Thm. 1.8] for an
efficient proof).

With this cooking receipt a large class of (sparse) moment problems,
Nichtnegativestellensätze, and Positivstellensätze can be generated, solved,
and efficiently proved. We think this makes it very useful for applications
and further theoretical investigations. $\circ$

##  6 Summary

In this work we review and deal with univariate sparse moment problems,
Positivstellensätzen, and Nichtnegativestellensätzen. We look at earlier
results and then move to the theory of T-systems. In the center are the works
of Karlin [20] and Karlin and Studden [23].

From 5.1.41 and 5.1.42 on $[a,b]$ we deduce a complete description of all
strictly positive sparse algebraic polynomials in 5.2.47. We also give the
sparse algebraic Nichtnegativestellensatz on $[a,b]$ in 5.2.53. With these
results we completely solve the sparse Hausdorff moment problem in 5.2.49,
5.2.51, and for the most general form in 5.2.52.

Following the extension by Karlin and Studden of 5.1.41 and 5.1.42 from
$[a,b]$ to $[0,\infty)$ we formulate the corresponding sparse algebraic
Positivstellensatz on $[0,\infty)$ in 5.3.58. Only the sparse algebraic
Positivstellensatz on $[0,\infty)$ is given since it already solves the sparse
Stieltjes moment problem in 5.3.59. The sparse algebraic
Nichtnegativstellensatz on $[0,\infty)$ can easily be derived like the sparse
Nichtnegativestellensatz on $[a,b]$ in 5.2.53.

We also give the general T-system Positivstellensatz on $\mathds{R}$ by Karlin
in 5.4.60. We give a general “cooking receipt” how other results in [23] (and
later literature) can be used to generate additional sparse algebraic
Positivstellensätze, Nichtnegativestellensätze, or to formulate and solve
sparse moment problems.

In this treatment we see the high value of the results in [23] which are
rarely used today. Especially the analytic treatment of the algebraic
questions seems unusual at first. However, we hope we convinced the reader
that this approach has at least in the univariate case (4.1.14) great value to
gain sparse Positivstellensätze, sparse Nichtnegativstellensätze, and
solutions to sparse moment problems.

## Funding

The author and this project are supported by the Deutsche
Forschungsgemeinschaft DFG with the grant DI-2780/2-1 and his research
fellowship at the Zukunfskolleg of the University of Konstanz, funded as part
of the Excellence Strategy of the German Federal and State Government.

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