Injectivity and surjectivity of the Stieltjes moment mapping in Gelfand-Shilov spaces

TL;DR

This paper investigates the conditions under which the Stieltjes moment mapping is injective or surjective within Gelfand-Shilov spaces, linking these properties to growth conditions of defining weight sequences.

Contribution

It provides a characterization of injectivity and surjectivity of the Stieltjes moment mapping in Gelfand-Shilov spaces based on weight sequence growth conditions.

Findings

Characterization of injectivity conditions

Characterization of surjectivity conditions

Analysis of a related moment problem at the origin

Abstract

The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.

Equations

$${\int }_0^{\infty }{x}^p\mathrm{d}F(x)=c_p,p\in \mathbb{N}=\{0,1,2,\dots \}.$$

$${\int }_0^{\infty }{x}^p\phi (x)\mathrm{d}x=c_p,p\in \mathbb{N},$$

$$\underset{p,q\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|x^p{\phi }^{(q)}(x)|}{h^{p+q}p!M_{p}q!A_q}<\infty$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|x^p{\phi }^{(q)}(x)|}{h^{p}p!M_p}<\infty ,\forall q\in \mathbb{N},$$

$$\displaystyle\sum_{p=0}^{\infty}\Big{(}\frac{M_{p}}{M_{p+1}}\Big{)}^{1/\gamma({\bf{M}})}=\infty,$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{x\in [0,1]}{\sup }\frac{|{\phi }^{(p)}(x)|}{h^{p}p!M_p}<\infty$$

$${\mu }_p^0(\phi )={\int }_0^1\frac{\phi (x)}{x^p}\mathrm{d}x,p\in \mathbb{N}.$$

$$M_j{M}_p\le M_{j+p},j,p\in \mathbb{N}.$$

$${\omega }_{\mathbf{M}}(t):=\underset{p\in \mathbb{N}}{\sup }\log \frac{t^p}{M_p},t>0,$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{z\in \mathbb{H}}{\sup }\frac{|f^{(p)}(z)|}{h^{p}p!M_p}<\infty .$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|f^{(p)}(x)|}{h^{p}p!M_p}<\infty .$$

$$f_p(x)=\underset{z\in \mathbb{H},z\to x}{\lim }{f}^{(p)}(z)\in \mathbb{C}$$

$$f(x)=\underset{z\in \mathbb{H},z\to x}{\lim }f(z),x\in \mathbb{R}.$$

$$f^{(p)}(x)=\underset{z\in \mathbb{H},z\to x}{\lim }{f}^{(p)}(z)$$

$$\underset{p\in \mathbb{N}}{\sup }\frac{|c_p|}{h^{p}p!M_p}<\infty .$$

$$\mathcal{B}:{\mathcal{A}}_{\mathbf{M}}(\mathbb{H})\to {\Lambda }_{\mathbf{M}}:f\to (f^{(p)}(0){)}_p,$$

$$\gamma (\mathbf{M}):=\sup \{\mu >0|(m_p/(p+1{)}^{\mu }{)}_{p}is\; almost\; increasing\}\in [0,\infty ]$$

$$({\gamma }_{\beta })\sum _{q=p}^{\infty }\frac{1}{(m_q{)}^{1/\beta }}\le \frac{C(p+1)}{(m_p{)}^{1/\beta }},p\in \mathbb{N}.$$

$$\gamma (\mathbf{M})=\sup \{\beta >0|\mathbf{M}\text{satisfies}({\gamma }_{\beta })\}.$$

$$\underset{p\in \mathbb{N}}{\sup }\frac{m_p}{p+1}\sum _{q=p}^{\infty }\frac{1}{m_q}<\infty ,$$

$$\omega (\mathbf{M}):=\underset{p\to \infty }{\mathrm{liminf}}\frac{\log (m_p)}{\log (p)}\in [0,\infty ],$$

$$\omega (\mathbf{M})$$

$$=\sup \{\mu >0|\sum _{p=0}^{\infty }\frac{1}{((p+1)m_p{)}^{1/(\mu +1)}}<\infty \}.$$

$$\sum _{p=0}^{\infty }\frac{1}{((p+1)m_p{)}^{1/2}}=\infty ,$$

$$\underset{p,q\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|x^p{\phi }^{(q)}(x)|}{h^{p+q}p!M_{p}q!A_q}<\infty .$$

$$\underset{q\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|{\phi }^{(q)}(x)|e^{{\omega }_{\hat{M}}(|x|/h)}}{h^{q}q!A_q}<\infty .$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{|x^p{\phi }^{(q)}(x)|}{h^{p}p!M_p}<\infty$$

$${\mathcal{S}}_{\mathbf{M}}^{\mathbf{A}}(0,\infty ):=\{\phi \in {\mathcal{S}}_{\mathbf{M}}^{\mathbf{A}}(\mathbb{R})|\mathrm{supp}\phi \subseteq [0,\infty )\}$$

$${\mathcal{S}}_{\mathbf{M}}(0,\infty ):=\{\phi \in {\mathcal{S}}_{\mathbf{M}}(\mathbb{R})|\mathrm{supp}\phi \subseteq [0,\infty )\}.$$

$$\mathcal{F}(\phi )(\xi )=\hat{\phi }(\xi )={\int }_{-\infty }^{\infty }\phi (x)e^{ix\xi }\mathrm{d}x,\phi \in L^1(\mathbb{R}).$$

$$\underset{x\in \mathbb{R}}{\sup }|x^p{\phi }^{(q)}(x)|\le C{h}^{p+q}p!M_{p}q!A_q,p,q\in \mathbb{N}.$$

Full text

1. Introduction

The moment problem, with its many variations and generalizations, has a long and rich tradition that goes back to the seminal work of Stieltjes [20]. In 1939, Boas [1] and Pólya [18] independently showed that, for every sequence $(c_{p})_{p=0}^{\infty}$ of complex numbers, there is a function $F$ of bounded variation such that

[TABLE]

This result was greatly improved by A. J. Durán [5] in 1989 who proved, in a constructive way, that, for every sequence $(c_{p})_{p\in\mathbb{N}}$ of complex numbers, the infinite system of linear equations

[TABLE]

admits a solution $\varphi\in\mathcal{S}(0,\infty)$, that is, $\varphi$ belongs to the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing smooth functions and $\operatorname{supp}\varphi\subseteq[0,\infty)$. We would like to point out that this result also follows from a short non-constructive argument via Eidelheit’s theorem [17, Thm. 26.27].

In this article, we study the (unrestricted) Stieltjes moment problem (1.1) in the context of Gelfand-Shilov spaces defined via weight sequences [7]; see [4, 13, 14] for earlier works in this direction. Namely, given two sequences of positive real numbers ${\bf{M}}=(M_{p})_{p\in\mathbb{N}}$ and ${\bf{A}}=(A_{p})_{p\in\mathbb{N}}$, we consider the spaces $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)$ and $\mathcal{S}_{{\bf{M}}}(0,\infty)$ consisting of all $\varphi\in\mathcal{S}(0,\infty)$ such that there exists $h>0$ with

[TABLE]

and

[TABLE]

respectively. Obviously, $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)\subset\mathcal{S}_{{\bf{M}}}(0,\infty)$. Now suppose that ${\bf{M}}$ is derivation closed, that is, $M_{p+1}\leq C_{0}H^{p+1}M_{p}$, $p\in\mathbb{N}$, for some $C_{0},H\geq 1$. Clearly, for every $\varphi\in\mathcal{S}_{{\bf{M}}}(0,\infty)$, its sequence of moments $(\mu_{p}(\varphi))_{p\in\mathbb{N}}=(\int_{0}^{\infty}x^{p}\varphi(x){\rm d}x)_{p\in\mathbb{N}}$ belongs to the sequence space $\Lambda_{{\bf{M}}}=\{(c_{p})_{p\in\mathbb{N}}\,|\,\exists h>0\,:\,\sup_{p\in\mathbb{N}}\frac{|c_{p}|}{h^{p}p!M_{p}}<\infty\}.$ It is then natural to ask about the surjectivity of the Stieltjes moment mapping $\mathcal{M}$, given by $\mathcal{M}(\varphi)=(\mu_{p}(\varphi))_{p\in\mathbb{N}}$, when defined on either $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)$ or $\mathcal{S}_{{\bf{M}}}(0,\infty)$ and with range $\Lambda_{{\bf{M}}}$. As a first result, following a technique of A. L. Durán and R. Estrada [6] that combines the Fourier transform with the Borel-Ritt theorem from asymptotic analysis, S.-Y. Chung, D. Kim and Y. Yeom [4, Thm. 3.1] proved the surjectivity of $\mathcal{M}$ for $\mathcal{S}_{{\bf{M}}}(0,\infty)$ with ${\bf{M}}=(p!^{\alpha-1})_{p\in\mathbb{N}}$ (the Gevrey sequence) whenever $\alpha>2$. Subsequently, A. Lastra and the third author [13] refined this result by obtaining linear continuous right inverses for the Stieltjes moment mapping between suitable Fréchet subspaces of $\mathcal{S}_{p!^{\alpha-1}}(0,\infty)$ and extended this result [14] to $\mathcal{S}_{{\bf{M}}}(0,\infty)$ for general strongly regular sequences ${\bf{M}}$, that is, sequences ${\bf{M}}$ that are log-convex, of moderate growth and strongly nonquasianalytic, whose growth index $\gamma({\bf{M}})$ is strictly greater than 1 (see Section 2 for the definition of these conditions and $\gamma({\bf{M}})$). Conversely, it was proven in [14] that if ${\bf{M}}$ is strongly regular, $\mathcal{M}:\mathcal{S}_{{\bf{M}}}(0,\infty)\to\Lambda_{{\bf{M}}}$ is surjective and

[TABLE]

then $\gamma({\bf{M}})>1$.

Our aim is to improve and complete these results by including the spaces $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)$ in our considerations, by dropping some hypotheses on ${\bf{M}}$, specially moderate growth and (1.2), and by also studying the injectivity of the Stieltjes moment mapping. Our key tools are: a better understanding of the meaning of the different growth conditions usually imposed on the sequence ${\bf{M}}$ and their expression in terms of indices of O-regular variation, as developed in [10]; the use of the Fourier transform in order to translate our problems into the corresponding ones for the asymptotic Borel mapping in certain ultraholomorphic classes on the upper half-plane; the enhanced information obtained in [11] about the injectivity and surjectivity of the asymptotic Borel mapping for sequences ${\bf{M}}$ subject to minimal conditions.

The paper is organized as follows. In the preliminary Section 2 we gather the main facts needed regarding sequences, ultraholomorphic classes, the asymptotic Borel mapping $\mathcal{B}$ and growth indices related to the injectivity and surjectivity of $\mathcal{B}$, Gelfand-Shilov spaces and the Laplace transform. Section 3 contains the main results. Firstly, in Theorem 3.4, we characterize the injectivity of the Stieltjes moment mapping $\mathcal{M}$ (defined on either $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)$ or $\mathcal{S}_{{\bf{M}}}(0,\infty)$) by an easy condition on the sequence ${\bf{M}}$, under minimal conditions on both ${\bf{M}}$ and ${\bf{A}}$. In Theorem 3.5 the surjectivity of $\mathcal{M}$ is characterized by the condition $\gamma({\bf{M}})>1$, for ${\bf{M}}$ log-convex and of moderate growth and ${\bf{A}}$ weakly log-convex and non-quasianalytic. In particular, this result significantly improves those in [14] and, moreover, extends to a general situation the statement of surjectivity of $\mathcal{M}$ in the case of the space $\mathcal{S}_{(p!^{\alpha-1})}^{(p!^{\beta-1})}(0,\infty)$, with $\alpha>2$ and $\beta>1$, that appeared (without proof) in [4, Thm. 3.3] and which is, up to the best of our knowledge, the only known result dealing with spaces of the type $\mathcal{S}_{{\bf{M}}}^{{\bf{A}}}(0,\infty)$. If moderate growth for ${\bf{M}}$ is substituted by the weaker condition of derivation closedness, we are only able to prove that $\gamma({\bf{M}})>1$ is necessary for the surjectivity of $\mathcal{M}$. We conclude this section by showing that the Stieltjes moment mapping is never bijective and that there exist strongly regular sequences for which $\mathcal{M}$ is neither injective nor surjective. The final Section 4 is devoted to the study of a related moment problem “at the origin”. More precisely, we consider the space $\mathcal{D}^{{\bf{M}}}(0,1)$ consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,1]$ such that there exists $h>0$ with

[TABLE]

and we define

[TABLE]

The study of the injectivity and surjectivity of the mapping \mathcal{M}^{0}:\mathcal{D}^{{{\bf{M}}}}(0,1)\rightarrow\Lambda_{{\mathchoice{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.93092pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.93092pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=3.34833pt\vrule height=0.0pt,width=4.49167pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.61203pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.61203pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=2.39166pt\vrule height=0.0pt,width=3.20833pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}}:\varphi\rightarrow(\mu^{0}_{p}(\varphi))_{p}, where {\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=9.16669pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=9.16669pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=4.58334pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}=(M_{p}/p!)_{p\in\mathbb{N}}, is reduced to the one of $\mathcal{M}$ via a suitable use of the Fourier transform.

2. preliminaries

2.1. Weight sequences

We set $\mathbb{N}=\{0,1,2,\dots\}$. Throughout this article ${\bf{M}}=(M_{p})_{p\in\mathbb{N}}$ will stand for a sequence of positive real numbers with $M_{0}=1$. We define $m_{p}=M_{p+1}/M_{p}$, $p\in\mathbb{N}$. The sequence ${\bf{M}}$ is said to be a weight sequence if $m_{p}\rightarrow\infty$ as $p\to\infty$. Furthermore, we set $\widehat{{\bf{M}}}=(p!M_{p})_{p\in\mathbb{N}}$ and {\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=9.16669pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=9.16669pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=4.58334pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}=(M_{p}/p!)_{p\in\mathbb{N}}. We shall use the following conditions on sequences ${\bf{M}}$:

• $\operatorname{(lc)}$

${\bf{M}}$ is log-convex if $M^{2}_{p}\leq M_{p-1}M_{p+1}$, $p\in\mathbb{Z}_{+}=\{1,2,\dots\}$.

• $\operatorname{(wlc)}$

${\bf{M}}$ is weakly log-convex if $\widehat{{\bf{M}}}$ satisfies $\operatorname{(lc)}$.

• $\operatorname{(dc)}$

${\bf{M}}$ is derivation closed if $M_{p+1}\leq C_{0}H^{p+1}M_{p}$, $p\in\mathbb{N}$, for some $C_{0},H\geq 1$.

• $\operatorname{(mg)}$

${\bf{M}}$ has moderate growth if $M_{p+q}\leq C_{0}H^{p+q}M_{p}M_{q}$, $p,q\in\mathbb{N}$, for some $C_{0},H\geq 1$.

• $\operatorname{(nq)}$

${\bf{M}}$ is non-quasianalytic if $\displaystyle\sum_{p=0}^{\infty}\frac{1}{(p+1)m_{p}}<\infty$.

• $\operatorname{(snq)}$

${\bf{M}}$ is strongly non-quasianalytic if $\displaystyle\sum_{q=p}^{\infty}\frac{1}{(q+1)m_{q}}\leq\frac{C}{m_{p}}$, $p\in\mathbb{N}$, for some $C>0$.

Remark 2.1*.*

All these properties are preserved when passing from ${\bf{M}}$ to $\widehat{{\bf{M}}}$. In particular, a sequence satisfying $\operatorname{(lc)}$ is also $\operatorname{(wlc)}$. However, only $\operatorname{(dc)}$ and $\operatorname{(mg)}$ are generally kept when going from ${\bf{M}}$ to

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=9.16669pt}$

$\textstyle{\bf{M}}$ .

A sequence ${\bf{M}}$ satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$ is easily proved to be a weight sequence. A sequence ${\bf{M}}$ is said to be strongly regular if it satisfies $\operatorname{(lc)}$, $\operatorname{(mg)}$ and $\operatorname{(snq)}$ (so, ${\bf{M}}$ is a weight sequence). The Gevrey sequence $(p!^{\alpha})_{p}$ ($\alpha>0$) is strongly regular.

In the classical work of H. Komatsu [12], the properties $\operatorname{(lc)}$, $\operatorname{(dc)}$ and $\operatorname{(mg)}$ are denoted by $(M.1)$, $(M.2)^{\prime}$ and $(M.2)$, respectively, while $\operatorname{(nq)}$ and $\operatorname{(snq)}$ for ${\bf{M}}$ are the same as properties $(M.3)^{\prime}$ and $(M.3)$ for $\widehat{{\bf{M}}}$, respectively.

For later use, we recall that $\operatorname{(lc)}$ (together with $M_{0}=1$) implies that

[TABLE]

Following Komatsu [12], the relation ${\bf{M}}\subset{\bf{N}}$ between two sequences means that there are $C,h>0$ such that $M_{p}\leq Ch^{p}N_{p}$ for all $p\in\mathbb{N}$. The associated function of ${\bf{M}}$ is defined as

[TABLE]

and $\omega_{{\bf{M}}}(0):=0$.

The following technical lemma shall be used later on.

Lemma 2.2**.**

Let ${\bf{M}}$ be a (weight) sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$. Then, there is a (weight) sequence ${\bf{N}}$ with ${\bf{N}}\subset{\bf{M}}$ satisfying $\operatorname{(wlc)}$, $\operatorname{(dc)}$ and $\operatorname{(nq)}$.

Proof.

Define $a_{p}=\min\{2^{p+1},(p+1)m_{p}\}$, $p\in\mathbb{N}$, and $N_{0}=1$; $N_{p}=\frac{1}{p!}\prod_{j=0}^{p-1}a_{j}$, $p\in\mathbb{Z}_{+}$. It is straightforward to check that ${{\bf{N}}}=(N_{p})_{p\in\mathbb{N}}$ satisfies all the requirements. ∎

2.2. Ultraholomorphic classes on the upper half-plane and the asymptotic Borel mapping

We write $\mathbb{H}$ for the open upper half-plane of the complex plane $\mathbb{C}$ and, given an open set $\Omega\subseteq\mathbb{C}$, we denote by $\mathcal{O}(\Omega)$ the space of holomorphic functions in $\Omega$. Let ${\bf{M}}$ be a weight sequence. For $h>0$ we define $\mathcal{A}_{{\bf{M}},h}(\mathbb{H})$ as the space consisting of all $f\in\mathcal{O}(\mathbb{H})$ such that

[TABLE]

We set $\mathcal{A}_{{\bf{M}}}(\mathbb{H})=\bigcup_{h>0}\mathcal{A}_{{\bf{M}},h}(\mathbb{H})$. Next, for $h>0$ we define $\mathcal{E}^{{\bf{M}},h}_{\infty}(\mathbb{R})$ as the space consisting of all $f\in C^{\infty}(\mathbb{R})$ such that

[TABLE]

We set $\mathcal{E}^{{\bf{M}}}_{\infty}(\mathbb{R})=\bigcup_{h>0}\mathcal{E}^{{\bf{M}},h}_{\infty}(\mathbb{R})$.

The following result is standard; it follows from the fact that the elements of $\mathcal{A}_{{\bf{M}},h}(\mathbb{H})$ together with all their derivatives are Lipschitz on $\mathbb{H}$.

Lemma 2.3**.**

Let ${\bf{M}}$ be a weight sequence and let $f\in\mathcal{A}_{{\bf{M}},h}(\mathbb{H})$ for some $h>0$. Then,

[TABLE]

exists for all $x\in\mathbb{R}$ and $p\in\mathbb{N}$. Moreover, $f_{0}\in C^{\infty}(\mathbb{R})$ and $(f_{0})^{(p)}=f_{p}$ for all $p\in\mathbb{N}$. Consequently, $f_{0}\in\mathcal{E}^{{\bf{M}},h}_{\infty}(\mathbb{R})$.

Remark 2.4*.*

Let ${\bf{M}}$ be a weight sequence and let $f\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$. In the sequel, we shall simply write

[TABLE]

Lemma 2.3 states that $f$ is continuous on $\overline{\mathbb{H}}$, $f_{|\mathbb{R}}\in\mathcal{E}^{{\bf{M}}}_{\infty}(\mathbb{R})$ and that

[TABLE]

for all $x\in\mathbb{R}$ and $p\in\mathbb{N}$.

Let ${\bf{M}}$ be a weight sequence. For $h>0$ we define $\Lambda_{{\bf{M}},h}$ as the space consisting of all sequences $(c_{p})_{p}\in\mathbb{C}^{\mathbb{N}}$ such that

[TABLE]

We set $\Lambda_{{\bf{M}}}=\bigcup_{h>0}\Lambda_{{\bf{M}},h}$. The asymptotic Borel mapping is defined as

[TABLE]

which is well-defined by Lemma 2.3 (see also Remark 2.4). For a fairly complete account on the injectivity and surjectivity of the asymptotic Borel mapping on various ultraholomorphic classes defined on arbitrary sectors we refer to [11]. There, two indices $\gamma({\bf{M}})$ and $\omega({\bf{M}})$, associated to the sequence ${\bf{M}}$, play a prominent role. In [8, Ch. 2] and [10, Sect. 3], the connections between these indices, the growth properties usually imposed on sequences, and the theory of O-regular variation, have been thoroughly studied. We summarize here some facts. The first index, introduced by V. Thilliez [21, Sect. 1.3] for strongly regular sequences, may be defined for any weight sequence ${\bf{M}}$ satisfying $\operatorname{(lc)}$ as

[TABLE]

(a sequence $(c_{p})_{p}$ is almost increasing if there exists $a>0$ such that $c_{p}\leq ac_{q}$ for all $q\geq p$). On the other hand, for $\beta>0$ we say that ${\bf{M}}$ satisfies $(\gamma_{\beta})$ if there is $C>0$ such that

[TABLE]

Then one has that

[TABLE]

Moreover, the following statements hold:

1. (i)

$\gamma({\bf{M}})>0$ if and only if ${\bf{M}}$ satisfies $\operatorname{(snq)}$. 2. (ii)

$\gamma({\bf{M}})>\beta$ if and only if ${\bf{M}}$ satisfies $(\gamma_{\beta})$.

The surjectivity of the asymptotic Borel mapping can be characterized as follows.

Theorem 2.5**.**

([11, Thm. 4.17])* Let ${\bf{M}}$ be a strongly regular weight sequence. Then, $\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$ is surjective if and only if*

[TABLE]

or, equivalently, $\gamma({\bf{M}})>1$.

The second index $\omega({\bf{M}})$ is given by

[TABLE]

and it turns out that

[TABLE]

Concerning the injectivity of the asymptotic Borel mapping, we have the next result.

Theorem 2.6**.**

([19, Thm. 12], [11, Thm. 3.4])* Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$. Then, $\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$ is injective if and only if*

[TABLE]

which in turn implies that $\omega({\bf{M}})\leq 1$.

Finally, we mention that if ${\bf{M}}$ is a weight sequence satisfying $\operatorname{(lc)}$, the asymptotic Borel mapping $\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$ is not bijective [11, Thm. 3.17].

2.3. Gelfand-Shilov spaces

Let ${\bf{M}}$ and ${\bf{A}}$ be weight sequences. For $h>0$ we define $\mathcal{S}_{{\bf{M}},h}^{{\bf{A}},h}(\mathbb{R})$ as the space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ such that

[TABLE]

Notice that $\varphi\in C^{\infty}(\mathbb{R})$ belongs to $\mathcal{S}_{{\bf{M}},h}^{{\bf{A}},h}(\mathbb{R})$ if and only if

[TABLE]

We set $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(\mathbb{R})=\bigcup_{h>0}\mathcal{S}_{{\bf{M}},h}^{{\bf{A}},h}(\mathbb{R})$. Analogously, we define $\mathcal{S}_{{\bf{M}},h}(\mathbb{R})$, $h>0$, as the space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ such that, for all $q\in\mathbb{N}$,

[TABLE]

and set $\mathcal{S}_{{\bf{M}}}(\mathbb{R})=\bigcup_{h>0}\mathcal{S}_{{\bf{M}},h}(\mathbb{R})$. As in the introduction, we define

[TABLE]

and

[TABLE]

Recall that $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)\subset\mathcal{S}_{{\bf{M}}}(0,\infty)$. Suppose that ${\bf{A}}$ satisfies $\operatorname{(wlc)}$, then $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$ is non-trivial if and only if ${\bf{A}}$ satisfies $\operatorname{(nq)}$, as follows from the Denjoy-Carleman theorem.

In the remainder of this subsection we determine the image of the spaces $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(\mathbb{R})$ and $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$ under the Fourier transform (cf. [7, Sect. IV.6]). We fix the constants in the Fourier transform as follows

[TABLE]

Proposition 2.7**.**

Let ${\bf{M}}$ and ${\bf{A}}$ be weight sequences satisfying $\operatorname{(wlc)}$ and $\operatorname{(dc)}$. Then, the Fourier transform is an isomorphism from $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(\mathbb{R})$ onto $\mathcal{S}^{{\bf{M}}}_{{\bf{A}}}(\mathbb{R})$.

Proof.

Since the Fourier transform is an isomorphism on the Schwartz space $\mathcal{S}(\mathbb{R})$ and $\mathcal{F}^{-1}(\varphi)(\xi)=(2\pi)^{-1}\mathcal{F}(\varphi)(-\xi)$ for all $\varphi\in\mathcal{S}(\mathbb{R})$, it suffices to show that $\mathcal{F}(\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(\mathbb{R}))\subseteq\mathcal{S}^{{\bf{M}}}_{{\bf{A}}}(\mathbb{R})$. Let $h\geq 1$ and $\varphi\in\mathcal{S}_{{\bf{M}},h}^{{\bf{A}},h}(\mathbb{R})$ be arbitrary. Choose $C>0$ such that

[TABLE]

Since ${\bf{M}}$ and ${\bf{A}}$ are weight sequences they are both increasing from some term on, and so there exists $D\geq 1$ such that $M_{j}\leq DM_{p}$ and $A_{j}\leq DA_{p}$ for all $j\leq p$. Hence,

[TABLE]

for all $p,q\in\mathbb{N}$. Therefore,

[TABLE]

for all $p,q\in\mathbb{N}$. ∎

In view of Proposition 2.7, the next result can be shown in a similar way as [3, Prop. 2.1].

Proposition 2.8**.**

Let ${\bf{M}}$ and ${\bf{A}}$ be weight sequences satisfying $\operatorname{(wlc)}$ and $\operatorname{(dc)}$. Let $\psi:\mathbb{R}\rightarrow\mathbb{C}$. Then, $\psi\in\mathcal{F}(\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty))$ if and only if $\psi\in\mathcal{S}^{{\bf{M}}}_{{\bf{A}}}(\mathbb{R})$ and there is $\Psi:\overline{\mathbb{H}}\rightarrow\mathbb{C}$ satisfying the following conditions:

• $(i)$

$\Psi_{|\mathbb{R}}=\psi$.

• $(ii)$

$\Psi$* is continuous on $\overline{\mathbb{H}}$ and analytic on $\mathbb{H}$.*

• $(iii)$

$\lim_{\zeta\in\overline{\mathbb{H}},\zeta\to\infty}\Psi(\zeta)=0$.

2.4. The Laplace transform

Let ${\bf{M}}$ be a weight sequence. We define $C_{{\bf{M}},h}[0,\infty)$ as the space consisting of all $\varphi\in C([0,\infty))$ such that

[TABLE]

or, in other words, such that

[TABLE]

We set $C_{{\bf{M}}}[0,\infty)=\bigcup_{h>0}C_{{\bf{M}},h}[0,\infty)$. The Laplace transform of $\varphi\in C_{{\bf{M}}}[0,\infty)$ is defined as

[TABLE]

Remark 2.9*.*

Let ${\bf{M}}$ and ${\bf{A}}$ be weight sequences. We may view $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$ and $\mathcal{S}_{{\bf{M}}}(0,\infty)$ as subspaces of $C_{{\bf{M}}}[0,\infty)$. Notice that $\mathcal{L}(\varphi)_{|\mathbb{R}}=\widehat{\varphi}$ for all $\varphi\in\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$.

Lemma 2.10**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(dc)}$. Then, the mapping $\mathcal{L}:C_{{\bf{M}}}[0,\infty)\rightarrow\mathcal{A}_{{\bf{M}}}(\mathbb{H})$ is well-defined and injective.

Proof.

The fact that $\mathcal{L}$ is well-defined follows along the same lines as the proof of Proposition 2.7. We now show that $\mathcal{L}$ is injective. Let $\varphi\in C_{{\bf{M}}}[0,\infty)$ be such that $\mathcal{L}(\varphi)\equiv 0$ on $\mathbb{H}$. Since $\mathcal{L}(\varphi)$ is continuous on $\overline{\mathbb{H}}$, we also have that $\mathcal{L}(\varphi)\equiv 0$ on $\mathbb{R}$. Define

[TABLE]

Then, $\widetilde{\varphi}\in L^{1}(\mathbb{R})$ and $\mathcal{F}(\widetilde{\varphi})=\mathcal{L}(\varphi)_{|\mathbb{R}}\equiv 0$. Since $\mathcal{F}$ is injective on $L^{1}(\mathbb{R})$, $\widetilde{\varphi}=0$ almost everywhere. As $\varphi$ is continuous on $[0,\infty)$, we may conclude that $\varphi\equiv 0$ on $[0,\infty)$. ∎

3. The Stieltjes moment problem in Gelfand-Shilov spaces

Let ${\bf{M}}$ be a weight sequence. The $p$-th moment, $p\in\mathbb{N}$, of an element $\varphi\in C_{{\bf{M}}}[0,\infty)$ is defined as

[TABLE]

If ${\bf{M}}$ satisfies $\operatorname{(dc)}$, then the Stieltjes moment mapping

[TABLE]

is well-defined. The goal of this section is to characterize the injectivity and surjectivity of the Stieltjes moment mapping on $C_{{\bf{M}}}[0,\infty)$ and its subspaces of type $\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$ and $\mathcal{S}_{{\bf{M}}}(0,\infty)$ in terms of the defining weight sequence ${\bf{M}}$. We employ the same idea as in [6], which was later also used in [3, 13, 14]. Namely, we shall reduce these problems to their counterparts for the asymptotic Borel mapping (Theorems 2.6 and 2.5) via the Laplace transform. In this regard, the following formula is fundamental

[TABLE]

In the next lemma we construct an auxiliary function that shall be frequently used throughout this section (compare with the function $G$ from [6]). We set $\mathbb{H}_{-1}=\{z\in\mathbb{C}\,|\,\Im m\,z>-1\}$.

Lemma 3.1**.**

Let ${\bf{A}}$ be a weight sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$. Then, there is $G\in\mathcal{O}(\mathbb{H}_{-1})$ satisfying the following conditions:

• $(i)$

$G$* does not vanish on $\mathbb{H}_{-1}$.*

• $(ii)$

$\displaystyle\sup_{z\in\mathbb{H}_{-1}}|G(z)|e^{\omega_{\widehat{A}}(|z|)}<\infty$.

• $(iii)$

$\displaystyle\sup_{p\in\mathbb{N}}\sup_{x\in\mathbb{R}}\frac{|G^{(p)}(x)|e^{\omega_{\widehat{A}}(|x|/2)}}{2^{p}p!}<\infty$.

The construction of the function $G$ from Lemma 3.1 is based on the following result.

Lemma 3.2**.**

([2, Lemma 2.2])* Let $\omega:[0,\infty)\rightarrow[0,\infty)$ be an increasing continuous function such that*

[TABLE]

and extend $\omega$ as an even function to the whole real line. Then, the Poisson transform of $\omega$ on $\mathbb{H}$ given by

[TABLE]

is harmonic and positive on $\mathbb{H}$ and satisfies

[TABLE]

Proof of Lemma 3.1.

Set $\omega=\omega_{\widehat{A}}(2\,\cdot)$. Since $A$ satisfies $\operatorname{(wlc)}$ and $\operatorname{(nq)}$, we have that [12, Lemma 4.1]

[TABLE]

Write $U=4P_{\omega}$ (cf. Lemma 3.2) and let $V$ be the harmonic conjugate of $U$ on $\mathbb{H}$. Define $G=e^{-(U(\,\cdot\,+i)+iV(\,\cdot\,+i))}$. It is clear that $G\in\mathcal{O}(\mathbb{H}_{-1})$ and that $(i)$ is satisfied. We now show $(ii)$ and $(iii)$.

$(ii)$: For $z\in\mathbb{H}_{-1}$ with $|z|\geq 2$ we have that $2|z+i|\geq|z|$ and, thus,

[TABLE]

For $z\in\mathbb{H}_{-1}$ with $|z|\leq 2$ we have that

[TABLE]

$(iii)$: By the Cauchy estimates and $(ii)$ there is $C>0$ such that

[TABLE]

for all $x\in\mathbb{R}$ and $p\in\mathbb{N}$. For $x\in\mathbb{R}$ with $|x|\geq 1$ we have that $|z|\geq|x|/2$ for all $z\in\mathbb{C}$ with $|z-x|\leq 1/2$. Hence,

[TABLE]

For $x\in\mathbb{R}$ with $|x|\leq 1$ we have that

[TABLE]

∎

Proposition 2.8 and Lemma 3.1 imply the following important lemma.

Lemma 3.3**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(dc)}$ and let ${\bf{A}}$ be a weight sequence satisfying $\operatorname{(wlc)}$, $\operatorname{(dc)}$ and $\operatorname{(nq)}$. Consider the function $G$ from Lemma 3.1. Then, $fG\in\mathcal{F}(\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty))$ for all $f\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$.

We are ready to study the injectivity and surjectivity of the Stieltjes moment mapping.

Theorem 3.4**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$ and let ${\bf{A}}$ be a weight sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$. Then, the following statements are equivalent:

• $(i)$

$\displaystyle\sum_{p=0}^{\infty}\frac{1}{((p+1)m_{p})^{1/2}}=\infty$.

• $(ii)$

$\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$* is injective.*

• $(iii)$

$\mathcal{M}:C_{{\bf{M}}}[0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is injective.*

• $(iv)$

$\mathcal{M}:\mathcal{S}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is injective.*

• $(v)$

$\mathcal{M}:\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is injective.*

Proof.

$(i)\Rightarrow(ii)$: By Theorem 2.6.

$(ii)\Rightarrow(iii)$: Let $\varphi\in C_{{\bf{M}}}[0,\infty)$ be such that $\mu_{p}(\varphi)=0$ for all $p\in\mathbb{N}$. By Lemma 2.10 we have that $\mathcal{L}(\varphi)\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$. Moreover, $\mathcal{L}(\varphi)^{(p)}(0)=i^{p}\mu_{p}(\varphi)=0$ for all $p\in\mathbb{N}$ and, thus, $\mathcal{L}(\varphi)\equiv 0$. Since $\mathcal{L}$ is injective (Lemma 2.10), we obtain that $\varphi\equiv 0$.

$(iii)\Rightarrow(iv)\Rightarrow(v)$: Obvious.

$(v)\Rightarrow(i)$: By Lemma 2.2 we may assume that ${\bf{A}}$ satisfies $\operatorname{(dc)}$. In view of Theorem 2.6 it suffices to show that $\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$ is injective. Let $f\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$ be such that $f^{(p)}(0)=0$ for all $p\in\mathbb{N}$. Consider the function $G$ from Lemma 3.1. By Lemma 3.3 we have that $fG=\widehat{\varphi}$ for some $\varphi\in\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$. Observe that

[TABLE]

Hence, $\varphi\equiv 0$ and, thus, $fG\equiv 0$. Since $G$ does not vanish (Lemma 3.1$(i)$), we obtain that $f\equiv 0$. ∎

Theorem 3.5**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$ and let ${\bf{A}}$ be a weight sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$. Then, the following statements are equivalent:

• $(i)$

$\mathcal{M}:\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is surjective.*

• $(ii)$

$\mathcal{M}:\mathcal{S}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is surjective.*

• $(iii)$

$\mathcal{M}:C_{{\bf{M}}}[0,\infty)\rightarrow\Lambda_{{\bf{M}}}$* is surjective.*

• $(iv)$

$\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$* is surjective.*

Each of the previous statements implies the next one:

• $(v)$

$\displaystyle\sup_{p\in\mathbb{N}}\frac{m_{p}}{p+1}\sum_{q=p}^{\infty}\frac{1}{m_{q}}<\infty$* or, equivalently, $\gamma({\bf{M}})>1$.*

If, in addition, ${\bf{M}}$ satisfies $\operatorname{(mg)}$, then all the previous statements are equivalent.

In the proof of Theorem 3.5 we shall use the following lemma (cf. [6]).

Lemma 3.6**.**

Let $(c_{p})_{p}\in\mathbb{C}^{\mathbb{N}}$ and let $G\in C^{\infty}((-\delta,\delta))$, for some $\delta>0$, such that $G(0)\neq 0$. Set

[TABLE]

Then,

[TABLE]

Proof.

Choose $0<\delta_{1}\leq\delta$ such that $G$ does not vanish on $(-\delta_{1},\delta_{1})$. By E. Borel’s theorem there is $f\in C^{\infty}((-\delta_{1},\delta_{1}))$ such that $f^{(p)}(0)=c_{p}$ for all $p\in\mathbb{N}$. Set $g=f/G\in C^{\infty}((-\delta_{1},\delta_{1}))$. Then,

[TABLE]

Hence,

[TABLE]

∎

Proof of Theorem 3.5.

We first prove the equivalence of the statements $(i)$ to $(iv)$.

$(i)\Rightarrow(ii)\Rightarrow(iii)$: Obvious.

$(iii)\Rightarrow(iv)$: Let $(c_{p})_{p}\in\Lambda_{{\bf{M}}}$ be arbitrary. Pick $\varphi\in C_{{\bf{M}}}[0,\infty)$ such that $\mu_{p}(\varphi)=(-i)^{p}c_{p}$ for all $p\in\mathbb{N}$. Then, $f=\mathcal{L}(\varphi)\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$ (Lemma 2.10) and $f^{(p)}(0)=i^{p}\mu_{p}(\varphi)=c_{p}$ for all $p\in\mathbb{N}$.

$(iv)\Rightarrow(i)$: By Lemma 2.2 we may assume that ${\bf{A}}$ satisfies $\operatorname{(dc)}$. Let $(c_{p})_{p}\in\Lambda_{{\bf{M}}}$ be arbitrary. Consider the function $G$ from Lemma 3.1. Set

[TABLE]

We claim that $(b_{p})_{p}\in\Lambda_{{\bf{M}}}$ (cf. [14, Prop. 6.4]). Indeed, choose $C,h>0$ such that $|c_{p}|\leq Ch^{p}p!M_{p}$ for all $p\in\mathbb{N}$. Next, since $1/G$ is holomorphic on a neighbourhood of the disk with center the origin and radius 1/2, there is $C^{\prime}>0$ such that $|(1/G)^{(p)}(0)|\leq C^{\prime}2^{p}p!$ for all $p\in\mathbb{N}$. Hence,

[TABLE]

where $D\geq 1$ is chosen so that $M_{j}\leq DM_{p}$ for all $j\leq p$. By assumption there is $f\in\mathcal{A}_{{\bf{M}}}(\mathbb{H})$ such that $f^{(p)}(0)=b_{p}$ for all $p\in\mathbb{N}$. We have that $fG=\widehat{\varphi}$ for some $\varphi\in\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)$ by Lemma 3.3. Finally, Lemma 3.6 implies that

[TABLE]

We now prove the statements related to $(v)$. The implication $(iv)\Rightarrow(v)$ follows directly from [11, Thm. 4.14$(i)$]. If, in addition, ${\bf{M}}$ satisfies $\operatorname{(mg)}$, condition $(v)$ implies that ${\bf{M}}$ satisfies $\operatorname{(snq)}$ as well (see Subsection 2.2), and so ${\bf{M}}$ is strongly regular. Then, Theorem 2.5 guarantees that $(iv)$ holds. ∎

Corollary 3.7**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$ and let ${\bf{A}}$ be a weight sequence satisfying $\operatorname{(wlc)}$ and $\operatorname{(nq)}$. Then, $\mathcal{M}:C_{{\bf{M}}}[0,\infty)\rightarrow\Lambda_{{\bf{M}}}$, $\mathcal{M}:\mathcal{S}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$ and $\mathcal{M}:\mathcal{S}^{{\bf{A}}}_{{\bf{M}}}(0,\infty)\rightarrow\Lambda_{{\bf{M}}}$ are never bijective.

Proof.

If any of the moment mappings were injective, we would have that $\sum_{p}((p+1)m_{p})^{-1/2}=\infty$ by Theorem 3.4. From Subsection 2.2 we deduce that $\omega({\bf{M}})\leq 1$, which in turn implies that $\gamma({\bf{M}})\leq 1$ because $\gamma({\bf{N}})\leq\omega({\bf{N}})$ for any weight sequence ${\bf{N}}$ satisfying $\operatorname{(lc)}$. Hence, $(v)$ from Theorem 3.5 is violated and therefore none of $(i)-(iii)$ from Theorem 3.5 can be satisfied. ∎

Example 3.8**.**

There exist strongly regular sequences for which the Stieltjes moment mapping is neither injective nor surjective. E.g., in [9, Example 4.18, Remark 4.19] (see also [8, Example 2.2.26]) the sequence ${\bf{M}}$ is defined via its sequence of quotients, $M_{p}=\prod_{j=0}^{p-1}m_{j}$, where

[TABLE]

and the sequence $(\delta_{k})_{k\in\mathbb{Z}_{+}}$ still has to be determined. Consider the sequences

[TABLE]

and choose $(\delta_{k})_{k}$ as follows:

[TABLE]

One can prove that ${\bf{M}}$ is strongly regular and that $\gamma({\bf{M}})=2<\omega({\bf{M}})=5/2$. Then, the sequence ${\bf{M}}^{1/2}:=(M^{1/2}_{p})_{p\in\mathbb{N}}$ is again strongly regular and $\gamma({\bf{M}}^{1/2})=1<5/4=\omega({\bf{M}}^{1/2})$. Hence, both the injectivity and surjectivity of the Stieltjes moment mapping are discarded.

Subsequently, in [8] (see also [10]), a general procedure has been designed to obtain strongly regular sequences with preassigned positive values of $\gamma({\bf{M}})$ and $\omega({\bf{M}})$. In particular, one can choose strongly regular sequences ${\bf{M}}$ with $\gamma({\bf{M}})\leq 1<\omega({\bf{M}})$ and thereby exclude both injectivity and surjectivity.

4. A moment problem at the origin

Let ${\bf{M}}$ be a weight sequence. For $h>0$ we define $\mathcal{D}^{{\bf{M}},h}(0,1)$ as the space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,1]$ such that

[TABLE]

We set $\mathcal{D}^{{{\bf{M}}}}(0,1)=\bigcup_{h>0}\mathcal{D}^{{\bf{M}},h}(0,1)$. Suppose that ${\bf{M}}$ satisfies $\operatorname{(wlc)}$, then $\mathcal{D}^{{{\bf{M}}}}(0,1)$ is non-trivial if and only if ${\bf{M}}$ satisfies $\operatorname{(nq)}$, as follows from the Denjoy-Carleman theorem. Notice that $\mathcal{D}^{{{\bf{M}}}}(0,1)\subset\mathcal{S}^{{\bf{M}}}_{{\bf{A}}}(0,\infty)$ for all weight sequences ${\bf{A}}$.

Lemma 4.1**.**

Let ${\bf{M}}$ be a weight sequence and let $\varphi\in\mathcal{D}^{{\bf{M}},h}(0,1)$ for some $h>0$. Then,

[TABLE]

for all $x\in[0,1]$ and $p\in\mathbb{N}$.

Proof.

Since $\varphi^{(j)}(0)=0$ for all $j\in\mathbb{N}$, Taylor’s theorem implies that

[TABLE]

for all $x\in[0,1]$ and $p\in\mathbb{N}$. ∎

Let ${\bf{M}}$ be a weight sequence. For $\varphi\in\mathcal{D}^{{{\bf{M}}}}(0,1)$ we define

[TABLE]

The mapping

[TABLE]

is well-defined by Lemma 4.1. The goal of this section is to characterize the injectivity and surjectivity of the mapping $\mathcal{M}^{0}$ in terms of the defining weight sequence ${\bf{M}}$. We shall reduce these problems to their counterparts for the Stieltjes moment mapping (Theorems 3.4 and 3.5) via the following lemma.

Lemma 4.2**.**

Let ${\bf{M}}$ be a weight sequence and let $\varphi\in\mathcal{D}^{{\bf{M}}}(0,1)$. Then,

[TABLE]

Proof.

For all $p\in\mathbb{N}$ we have that

[TABLE]

∎

We are ready to characterize the injectivity and surjectivity of the mapping $\mathcal{M}^{0}$.

Theorem 4.3**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$, $\operatorname{(dc)}$ and $\operatorname{(nq)}$. Then, the following statements are equivalent:

• $(i)$

$\displaystyle\sum_{p=0}^{\infty}\frac{1}{((p+1)m_{p})^{1/2}}=\infty$.

• $(ii)$

$\mathcal{M}:\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))\rightarrow\Lambda_{{\bf{M}}}$* is injective.*

• $(iii)$

\mathcal{M}^{0}:\mathcal{D}^{{{\bf{M}}}}(0,1)\rightarrow\Lambda_{{\mathchoice{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.93092pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.93092pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=3.34833pt\vrule height=0.0pt,width=4.49167pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.61203pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.61203pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=2.39166pt\vrule height=0.0pt,width=3.20833pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}}* is injective.*

Remark 4.4*.*

If we assume that ${\bf{M}}$ satisfies $\operatorname{(lc)}$ and it does not satisfy $\operatorname{(nq)}$, then $\mathcal{D}^{{{\bf{M}}}}(0,1)$ is trivial and $\sum_{p=0}^{\infty}((p+1)m_{p})^{-1}=\infty$, and so the three previous statements clearly hold true. This justifies the hypothesis $\operatorname{(nq)}$ in Theorem 4.3, while condition $\operatorname{(dc)}$ is needed in order to apply our previous results about the moment mapping $\mathcal{M}$.

The proof of Theorem 4.3 is based on the next result.

Proposition 4.5**.**

Let $\varphi\in L^{1}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,\infty)$. If $\widehat{\varphi}$ vanishes on a subset of $\mathbb{R}$ with positive Lebesgue measure, then $\varphi=0$ almost everywhere.

Proposition 4.5 follows directly from the Lusin-Privalov theorem, which we include here for the reader’s convenience.

Theorem 4.6** (of Lusin and Privalov [15]).**

Let $F\in\mathcal{O}(\mathbb{H})$ and suppose that there exists a set $A\subset\mathbb{R}$ with positive Lebesgue measure such that, for every $x\in A$ and $0<\delta<1$, it holds that

[TABLE]

where $S_{x,\delta}\subset\mathbb{H}$ is the sector with vertex at $x$, vertical bisecting direction and opening $\pi\delta$. Then, $F\equiv 0$ on $\mathbb{H}$.

Proof of Proposition 4.3.

$(i)\Rightarrow(ii)$: Follows directly from Theorem 3.4, since it is clear that $\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))\subset C_{{\bf{M}}}[0,\infty)$.

$(ii)\Rightarrow(i)$: By Theorem 3.4 it suffices to show that $\mathcal{M}:C_{{\bf{M}}}[0,\infty)\rightarrow\Lambda_{{\bf{M}}}$ is injective. Let $\varphi\in C_{{\bf{M}}}[0,\infty)$ be such that $\mu_{p}(\varphi)=0$ for all $p\in\mathbb{N}$. Set

[TABLE]

Next, by condition $\operatorname{(nq)}$, we can choose $\psi\in\mathcal{D}^{{{\bf{M}}}}(0,1)$ such that $\psi(x)\neq 0$ for all $x\in(0,1)$ . Then,

[TABLE]

as follows from Lemmas 2.3 and 2.10. Moreover, we have that

[TABLE]

Hence, $\widetilde{\varphi}\ast\widehat{\psi}\equiv 0$ and, thus, also $\mathcal{F}^{-1}(\widetilde{\varphi}\ast\widehat{\psi})=\mathcal{L}(\widetilde{\varphi})_{|\mathbb{R}}(-\,\cdot)\cdot\psi\equiv 0$. Since $\psi(x)\neq 0$ for all $x\in(0,1)$, we obtain that $\mathcal{F}(\widetilde{\varphi})(\xi)=\mathcal{L}(\varphi)_{|\mathbb{R}}(\xi)=0$ for all $\xi\in(-1,0)$. Proposition 4.5 yields that $\widetilde{\varphi}=0$ almost everywhere. As $\varphi$ is continuous on $[0,\infty)$, we may conclude that $\varphi\equiv 0$ on $[0,\infty)$.

$(ii)\Rightarrow(iii)$: Follows directly from Lemma 4.2.

$(iii)\Rightarrow(ii)$: Let $\varphi\in\mathcal{D}^{{{\bf{M}}}}(0,1)$ be such that $\mu_{p}(\widehat{\varphi})=0$ for all $p\in\mathbb{N}$. By Lemma 4.2 we have that $\mu^{0}_{p}(\varphi)=0$ for all $p\in\mathbb{Z}_{+}$. Condition $\operatorname{(dc)}$ implies that $\varphi^{\prime}\in\mathcal{D}^{{{\bf{M}}}}(0,1)$. Moreover, we have that $\mu^{0}_{0}(\varphi^{\prime})=\int_{0}^{1}\varphi^{\prime}(x){\rm d}x=0$ and

[TABLE]

Hence, $\varphi^{\prime}\equiv 0$ and, since $\varphi$ is compactly supported, we obtain that $\varphi\equiv 0$. ∎

Theorem 4.7**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$. Then, the following statements are equivalent:

• $(i)$

\mathcal{M}^{0}:\mathcal{D}^{{{\bf{M}}}}(0,1)\rightarrow\Lambda_{{\mathchoice{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.93092pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.93092pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=3.34833pt\vrule height=0.0pt,width=4.49167pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.61203pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.61203pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=2.39166pt\vrule height=0.0pt,width=3.20833pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}}* is surjective.*

• $(ii)$

$\mathcal{M}:\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))\rightarrow\Lambda_{{\bf{M}}}$* is surjective.*

Each of the previous statements implies the next one:

• $(iii)$

$\displaystyle\sup_{p\in\mathbb{N}}\frac{m_{p}}{p+1}\sum_{q=p}^{\infty}\frac{1}{m_{q}}<\infty$* or, equivalently, $\gamma({\bf{M}})>1$.*

If, in addition, ${\bf{M}}$ satisfies $\operatorname{(mg)}$, then all the previous statements are equivalent.

We need a lemma in preparation of the proof of Theorem 4.7.

Lemma 4.8**.**

Let ${\bf{M}}$ be a weight sequence and let $\chi\in C_{{\bf{M}}}[0,\infty)$ be such that $\mu_{0}(\chi)\neq 0$. Define $G=\mathcal{L}(\chi)_{|\mathbb{R}}$ and notice that $G(0)=\mu_{0}(\chi)\neq 0$. For $(c_{p})_{p}\in\mathbb{C}^{\mathbb{N}}$ we set

[TABLE]

Then,

[TABLE]

Proof.

Lemma 3.6 implies that

[TABLE]

∎

Proof of Theorem 4.7.

We first prove the equivalence of $(i)$ and $(ii)$.

$(i)\Rightarrow(ii)$: Follows directly from Lemma 4.2.

$(ii)\Rightarrow(i)$: Let (c_{p})_{p}\in\Lambda_{{\mathchoice{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.93092pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.93092pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=3.34833pt\vrule height=0.0pt,width=4.49167pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.61203pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.61203pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=2.39166pt\vrule height=0.0pt,width=3.20833pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}} be arbitrary. Set $b_{p}=i^{p+1}(p-1)!c_{p}$, $p\in\mathbb{N}$. Then, $(b_{p})_{p}\in\Lambda_{{\bf{M}}}$. Choose $\varphi\in\mathcal{D}^{{{\bf{M}}}}(0,1)$ such that $\mu_{p}(\widehat{\varphi})=b_{p}$ for all $p\in\mathbb{N}$. Consider the function $\varphi^{\prime}$, which belongs to $\mathcal{D}^{{{\bf{M}}}}(0,1)$ because of $\operatorname{(dc)}$. Then, the computation in (4.1) and Lemma 4.2 imply that

[TABLE]

For the second part of the theorem, observe that, since $\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))\subset C_{{\bf{M}}}[0,\infty)$, the implication $(ii)\Rightarrow(iii)$ follows directly from $(iii)\Rightarrow(v)$ in Theorem 3.5.

Finally, if ${\bf{M}}$ additionally satisfies $\operatorname{(mg)}$ and we depart from $(iii)$, as before we first deduce that ${\bf{M}}$ is strongly regular and, thus, satisfies $\operatorname{(nq)}$. Let $(c_{p})_{p}\in\Lambda_{{\bf{M}}}$ be arbitrary. Choose $\psi\in\mathcal{D}^{{{\bf{M}}}}(0,1)$ such that $\mu_{1}^{0}(\psi)\neq 0$. Define $\chi=\widehat{\psi}_{|[0,\infty)}\in C_{{\bf{M}}}[0,\infty)$ and $G=\mathcal{L}(\chi)_{|\mathbb{R}}$. Notice that, by Lemma 4.2, we have that $G(0)=\mu_{0}(\chi)=i\mu_{1}^{0}(\psi)\neq 0$. Set

[TABLE]

We claim that $(b_{p})_{p}\in\Lambda_{{\bf{M}}}$. Indeed, choose $C,h>0$ such that $|c_{p}|\leq Ch^{p}p!M_{p}$ for all $p\in\mathbb{N}$. Next, since $G\in\mathcal{E}^{{\bf{M}}}_{\infty}(\mathbb{R})$ (Lemmas 2.3 and 2.10), $G(0)\neq 0$ and ${\bf{M}}$ satisfies $\operatorname{(lc)}$, a classical result of Malliavin on the inverse-closedness of algebras of ultradifferentiable functions [16, p. 185, 4.1] implies that there are $C^{\prime},k>0$ such that $|(1/G)^{(p)}(0)|\leq C^{\prime}k^{p}p!M_{p}$ for all $p\in\mathbb{N}$. Hence,

[TABLE]

where we have used (2.1). By Theorem 3.5, part $(v)\Rightarrow(iii)$, there is $\varphi\in C_{{\bf{M}}}[0,\infty)$ such that $\mu_{p}(\varphi)=b_{p}$ for all $p\in\mathbb{N}$. Set

[TABLE]

Then, $\widetilde{\varphi}\ast\widehat{\psi}\in\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))$ (cf. the proof of Proposition 4.3). Finally, Lemma 4.8 implies that

[TABLE]

and so $(ii)$ holds. ∎

Corollary 4.9**.**

Let ${\bf{M}}$ be a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$. Then, $\mathcal{M}:\mathcal{F}(\mathcal{D}^{{{\bf{M}}}}(0,1))\rightarrow\Lambda_{{\bf{M}}}$ and \mathcal{M}^{0}:\mathcal{D}^{{{\bf{M}}}}(0,1)\rightarrow\Lambda_{{\mathchoice{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\displaystyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=6.41667pt}}}}}\cr\hbox{\textstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.93092pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.93092pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=3.34833pt\vrule height=0.0pt,width=4.49167pt}}}}}\cr\hbox{\scriptstyle{\bf{M}}}}}}{{\ooalign{\hbox{\raise 5.61203pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.61203pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=2.39166pt\vrule height=0.0pt,width=3.20833pt}}}}}\cr\hbox{\scriptscriptstyle{\bf{M}}}}}}}} are never bijective.

Proof.

In view of Theorems 4.3 and 4.7, this can be shown in exactly the same way as Corollary 3.7. ∎

Remark 4.10*.*

A careful inspection of the proofs of Theorems 3.5 and 4.7 shows that, as long as ${\bf{M}}$ is a weight sequence satisfying $\operatorname{(lc)}$ and $\operatorname{(dc)}$, the surjectivity of the Borel mapping $\mathcal{B}:\mathcal{A}_{{\bf{M}}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}}$ implies, not only $\gamma({\bf{M}})>1$, but also the surjectivity of all the moment mappings considered in both statements. Although the condition $\operatorname{(mg)}$, combined with $\operatorname{(lc)}$ and $\gamma({\bf{M}})>1$, allows one to prove the surjectivity of $\mathcal{B}$ (see Theorem 2.5), in some cases where $\operatorname{(mg)}$ fails one can still show that $\mathcal{B}$ is surjective. A very classical example is that of the so-called $q$-Gevrey sequences, ${\bf{M}}_{q}=(q^{p^{2}})_{p\in\mathbb{N}}$, where $q>1$. These sequences satisfy $\operatorname{(lc)}$, $\operatorname{(dc)}$ and $\gamma({\bf{M}}_{q})>1$ (indeed, $\gamma({\bf{M}}_{q})=\infty$) but not $\operatorname{(mg)}$. One can prove (see [21, Subsect. 3.3] for some hints and references) that $\mathcal{B}:\mathcal{A}_{{\bf{M}}_{q}}(\mathbb{H})\rightarrow\Lambda_{{\bf{M}}_{q}}$ is surjective and so the previous considerations apply to this case.

Acknowledgements: The first author is supported by FWO-Vlaanderen, via the postdoctoral grant 12T0519N. The last two authors are partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under the project MTM2016-77642-C2-1-P.

The authors wish to express their gratitude to Prof. Manuel Núñez, from the Universidad de Valladolid (Spain), for making them aware of Theorem 4.6.