Integral representations, extension theorems and walks through dimensions under radial exponential convexity

Abstract

We consider the class of radial exponentially convex functions defined over n-dimensional balls with finite or infinite radii. We provide characterization theorems for these classes, as well as Rudin’s type extension theorems for radial exponentially convex functions defined over n-dimensional balls into radial exponentially convex functions defined over the whole n-dimensional Euclidean space. We furthermore establish inversion theorems for the measures, termed here n-Nussbaum measures, associated with integral representations of radial exponentially convex functions. This in turn allows obtaining recurrence relations between 1-Nussbaum measures and n-Nussbaum measures for a given integer n greater than 1. We also provide a up to now unknown catalogue of radial exponentially convex functions and associated n-Nussbaum measures. We finally turn our attention into componentwise radial exponential convexity over product spaces, with a Rudin extension result and analytical examples of exponentially convex functions and associated Nussbaum measures. As a byproduct, we obtain a parametric model of nonseparable stationary space-time covariance functions that do not belong to the well-known Gneiting class.

1 Introduction

Exponentially convex functions have a long history that can be traced back to Bernstein (1929), Widder (1934), Loève (1946), Devinatz (1955) and Nussbaum (1972), and play an important role in the literature on complex and harmonic analysis, with many applications to several branches of applied sciences (Ehm et al. 2003). In probability theory and stochastic processes, exponential convexity is connected with locally stationary covariance functions in ${\mathbb {R}}^n$, which have been known by both statistical and mathematical communities since the early works by Silverman (1957, 1959) and inspired a large literature on reducibility problems, see Perrin and Senoussi (2000), Genton and Perrin (2004) and Porcu et al. (2020) amongst others. Specifically, let $\{Z(\textbf{x}), \; \textbf{x} \in {\mathbb {R}}^n \}$ be a random field with a locally stationary covariance function, C. Then, it is true that

$$\begin{aligned} C(\varvec{x},\varvec{y}) = C_1 \left( \varvec{y}- \varvec{x}\right) K \left( \varvec{x}+ \varvec{y}\right) , \qquad \varvec{x},\varvec{y}\in {\mathbb {R}}^n, \end{aligned}$$

(1)

Here, the function $(\varvec{x},\varvec{y}) \mapsto C_1(\varvec{y}-\varvec{x})$ is positive definite, while $K: {\mathbb {R}}^n \rightarrow {\mathbb {R}}$ is a strictly positive function that must be chosen in such a way that the product $C_1(\cdot )K(\cdot )$ is a positive definite function in ${\mathbb {R}}^n$. This is guaranteed if K is exponentially convex, i.e., if $(\varvec{x},\varvec{y}) \mapsto K \left( \varvec{x}+ \varvec{y}\right)$ with $(\varvec{x},\varvec{y}) \in {\mathbb {R}}^n \times {\mathbb {R}}^n$, is positive definite (Berg et al. 1984).

Widder (1934) and Devinatz (1955) proved that, for $A \subseteq {\mathbb {R}}^n$, $K: A \rightarrow {\mathbb {R}}$ is exponentially convex if and only if

$$\begin{aligned} K(\varvec{x}) = \int _{{\mathbb {R}}^n} \textrm{e}^{\langle \varvec{x}, \varvec{\omega }\rangle _n} \textrm{d} F(\varvec{\omega }), \qquad \varvec{x}\in A, \end{aligned}$$

(2)

where F is a nonnegative and finite measure such that the above integral converges for all $\varvec{x}\in {\mathbb {R}}^n$, and $\langle \cdot , \cdot \rangle _n$ is the inner product in ${\mathbb {R}}^n$. Hence, K is the bilateral Laplace transform of F (Widder 1934).

Ehm et al. (2003) provided a nice relation between positive definite functions and exponentially convex functions in ${\mathbb {R}}^n$. Specifically, they proved that, for an exponentially convex function K, the function ${(\varvec{x},\varvec{y}) \mapsto C(\varvec{x},\varvec{y})=K(\textsf{i}(\varvec{x}-\varvec{y}))}$, with $\textsf{i}$ denoting the unit complex number, is positive definite in ${{\mathbb {R}}^n \times {\mathbb {R}}^n}$. Conversely, they proved that, for an entire function C in ${\mathbb {R}}^n$ such that $(\varvec{x},\varvec{y}) \mapsto C(\varvec{x}-\varvec{y})$ is positive definite in ${\mathbb {R}}^n \times {\mathbb {R}}^n$, the function $\varvec{x}\mapsto K(\varvec{x}) = C(-\textsf{i} \varvec{x})$ is exponentially convex. This result can also be established by viewing $C/C(\varvec{0})$ as the characteristic function of a n-dimensional random vector and using Theorem 6.1.4 in Linnik and Ostrovskii (1977) to express K under the form (2).

This paper focuses on exponentially convex functions K that are radially symmetric in ${\mathbb {R}}^n$, or generalizations of them that will be discussed subsequently.

1.1 The problems

The current literature is minimal, if not elusive, with respect to the following:

• 1. Let K be exponentially convex over $A \subset {\mathbb {R}}^n$. Is it true that there exists at least a function $\widetilde{K}: {\mathbb {R}}^n \rightarrow {\mathbb {R}}$ that is exponentially convex and that is identically equal to K over A? Such an extension problem has been well known to the mathematical, probabilistic and statistical communities since the early fifties for the case of characteristic functions, being normalized versions of the function $C_1$ in Eq. (1), after the tour de force by Krein (1940), and subsequently by Rudin (1963, 1970). Later important contributions to this problem can be found in Sasvári (1994, 2006, 2013), Gneiting and Sasvári (1999) and Porcu et al. (2023). Up to now, the extension of an exponentially convex function defined over a subset of the n-dimensional Euclidean space has not been considered; we do not know whether such an extension exists and whether it is unique or not.

• 2. There is little understanding about the measures that are associated with the integral representation of radial exponentially convex functions. An attempt to describe spectral measures associated with exponentially convex functions is provided in Silverman (1957), but the results are unfortunately incorrect (see the subsequent Sect. 3.2 for technical details) and partially amended in Silverman (1959), with the addition of extremely restrictive technical conditions to ensure that such an inversion is feasible.

• 3. Relations between isotropic spectral measures have been under the radar of the mathematical and statistical communities for a long time. In particular, “dimension walks” based on relations between isotropic spectral measures on different Euclidean spaces have been extensively studied for radial positive definite functions, see Matheron (1965); Eaton (1981); Wendland (1995); Schaback and Wu (1996); Gneiting (2002a); Daley and Porcu (2014). The importance of such operators is witnessed by their role for the construction of compactly supported covariance functions for large data sets (Gneiting 2002b) or of covariance functions for random fields defined over balls embedded in ${\mathbb {R}}^n$ (Porcu et al. 2023, and references therein). Similar operators that relate pairs of radial exponentially convex functions defined over n and $n'$-dimensional balls have not been considered so far.

• 4. Exponentially convex functions that are componentwise radial over product spaces have not been considered. It is currently unknown how to attain a characterization theorem for such a class. Analogously, extension theorems on the style of Porcu et al. (2023) have never been considered. The motivation for digging into such classes comes from space-time statistics. Let $A \subset {\mathbb {R}}^n$ and $B \subset {\mathbb {R}}$. Positive definite functions that are radial in A and symmetric in B are widely used as space-time covariance functions, and the literature for this subject is ubiquitous (see Chen et al. 2021 and Porcu et al. 2021 for a thorough review). Locally stationary space-time covariance functions have not been studied so far, despite the fact that local stationarity is a fundamental ingredient in the study of natural processes that evolve over space and time. A trivial solution to this problem would be to use the fact that positive definite functions are a convex cone that is closed under the topology of finite measures. Hence, the product of a radial exponentially convex function in A with a symmetric exponentially convex function in B would provide an exponentially convex function in $A \times B$. Such a construction is called separable in spatial statistics, and it is considered suboptimal for as many reasons as those purported in Porcu et al. (2021).

1.2 Our contributions

This paper provides some effort to fill the above gaps. In particular,

• (a) We provide representation theorems for exponentially convex functions that are radial over ${\mathbb {R}}^n$, or over a ball ${\mathbb {B}}_n(r)$ with radii r embedded in ${\mathbb {R}}^n$. Additionally, we provide an extension theorem of the Rudin type (Rudin 1970) for radial exponentially convex functions from ${\mathbb {B}}_n(r)$ to ${\mathbb {R}}^n$, and we prove that such an extension is unique. This provides solutions to Problem 1 above.

• (b) We provide inversion theorems that allow obtaining measures related to the integral representation of radial exponentially convex functions in ${\mathbb {R}}^n$, and refer to them as n-Nussbaum measures. Such measures are inspected in detail, with emphasis on recursive formulae that allow attaining a 1-Nussbaum measure from a given n-Nussbaum measure. We also provide a catalogue of radial exponentially convex functions, together with their n-Nussbaum measures, that were unknown up to now. All this provides solutions to Problems 2 and 3 above.

• (c) We consider exponentially convex functions defined over product spaces, where the function is radial for every argument in the respective space. We provide an extension theorem of the Rudin type for such product spaces, as well as examples of nonseparable exponentially convex functions, where nonseparability means that the proposed structure do not factor into the product of separate radial exponentially convex functions over their respective spaces.

The remainder of the paper is as follows. Section 2 provides the necessary mathematical background. Section 3 contains our novel results. Succinct concluding remarks (Sect. 4) close the paper.

2 Background material

Let n be a positive integer, r a positive real number and ${\mathbb {B}}_n(r)$ the open ball of radius r in ${\mathbb {R}}^n$, defined as

$$\begin{aligned} {\mathbb {B}}_n(r)=\{ \varvec{x}\in {\mathbb {R}}^n: \Vert \varvec{x}\Vert _n < r \}, \end{aligned}$$

with $\Vert \cdot \Vert _n$ denoting the Euclidean norm in ${\mathbb {R}}^n$. By abuse of notation, we define ${\mathbb {B}}_n(\infty )$ as the Euclidean space ${\mathbb {R}}^n$.

For $r \in (0,\infty ]$, we call $\Phi _n(r)$ the class of continuous functions $\varphi : [0,r) \rightarrow {\mathbb {R}}$ such that $(\varvec{x},\varvec{y}) \mapsto \varphi (\Vert \varvec{y}-\varvec{x}\Vert _n)$ is positive definite over ${\mathbb {B}}_n(r/2)$, that is

$$\begin{aligned} \sum _{i=1}^N \sum _{j=1}^N a_i \, {\overline{a}}_j \, \varphi \left( \Vert \varvec{x}_i - \varvec{x}_j \Vert _n\right) \ge 0, \end{aligned}$$

for every $\{a_i \}_{i=1}^N \subset {{\mathbb {C}}}$ and $\{ \varvec{x}_i \}_{i=1}^N \subset {\mathbb {B}}_n(r/2)$. Here, ${\overline{a}}$ stands for the complex conjugate of a. Such functions are called radial positive definite, radially symmetric positive definite or isotropic positive definite over ${\mathbb {B}}_n(r)$ (Daley and Porcu 2014). Schoenberg’s theorem (Schoenberg 1938) proves that a mapping $\varphi$ belongs to $\Phi _n(\infty )$ if and only if

$$\begin{aligned} \varphi (x) = \int _{0}^{\infty } \Omega _n( x u) \textrm{d} \nu _n(u), \qquad x \in [0,\infty ), \end{aligned}$$

(3)

where $\nu _n$ is a unique nonnegative and finite measure (termed n-Schoenberg measure by Daley and Porcu (2014)) and $\Omega _n$ (called n-Schoenberg kernel therein) is defined in the complex plane as

$$\begin{aligned} \begin{aligned} {\Omega }_n(x)&= 1+\Gamma \left( \frac{n}{2}\right) \sum _{k=1}^{\infty } \frac{(-1)^k}{\Gamma (k+\frac{n}{2}) k!} \left( \frac{x}{2}\right) ^{2k} \\&= {\left\{ \begin{array}{ll} \Gamma \left( \frac{n}{2}\right) \left( \frac{2}{x}\right) ^{(n-2)/2} J_{(n-2)/2}(x), \qquad x \in {\mathbb {C}} \setminus \{0\}\\ 1, \qquad x = 0, \end{array}\right. } \end{aligned} \end{aligned}$$

(4)

with $\Gamma$ the gamma function and $J_{(n-2)/2}$ the Bessel function of the first kind of order $(n-2)/2$ (Olver et al. 2010, formula 10.2.2).

For $r \in (0,\infty ]$, a continuous function $\psi : [0,r) \rightarrow {\mathbb {R}}$ is said to be radial exponentially convex over ${\mathbb {B}}_n(r)$ if

$$\begin{aligned} \sum _{i=1}^N \sum _{j=1}^N a_i {\overline{a}}_j \psi \left( \Vert \varvec{x}_i + \varvec{x}_j \Vert _n\right) \ge 0, \end{aligned}$$

for every $\{a_i \}_{i=1}^N \subset {\mathbb {C}}$ and $\{ \varvec{x}_i \}_{i=1}^N \subset {\mathbb {B}}_n(r/2)$. We shall write $\text {EC}_n(r)$ for the class of radial exponentially convex functions over ${\mathbb {B}}_n(r)$. Clearly, the following inclusion relation is strict:

$$\begin{aligned} \text {EC}_1(r) \supset \text {EC}_2(r) \supset \cdots \supset \text {EC}_n(r)\supset \cdots \supset \text {EC}_{\infty }(r) =: \bigcap _{n \ge 1} \text {EC}_n(r). \end{aligned}$$

Nussbaum (1972, Proposition 2) proved that $\psi$ belongs to $\text {EC}_n(r)$ if and only if it has a representation of the form

$$\begin{aligned} \psi (x) = \int _{0}^{\infty } \Omega _n(\textsf{i} x u) \textrm{d} \gamma _n(u), \qquad x \in [0,r), \end{aligned}$$

(5)

where $\Omega _n$ has been defined in (4) and $\gamma _n$ is a nonnegative and finite Radon measure on $[0,\infty )$, that is, $\gamma _n$ is finite and inner regular on open sets. This in turn proves, in concert with a Schoenberg-type convergence argument as in Schoenberg (1938), that, for a given $r>0$, $\psi$ belongs to the class $\text {EC}_{\infty }(r)$ if and only if it is written as (Nussbaum 1972, Theorem 2)

$$\begin{aligned} \psi (x) = \int _{0}^{\infty } \textrm{e}^{x^2 u} \textrm{d} \gamma _n(u), \qquad x \in [0,r), \end{aligned}$$

i.e., $\psi \in \text {EC}_{\infty }(r)$ if and only if $\psi (\sqrt{\cdot })$ is absolutely monotonic on $[0,r^2)$ (McMillan 1954). The measure $\gamma _n$ in (5) is actually unique, as will be seen in Proposition 2 hereinafter, and will be referred to as the n-Nussbaum measure of $\psi$.

Finally, we say that $\psi :[0,r) \rightarrow {\mathbb {R}}$ is an entire function to mean that it has an entire extension to the complex plane ${\mathbb {C}}$.

3 Digging into radial exponential convexity

3.1 The class $\text {EC}_n(r)$

We define the n-Nussbaum kernel in the complex plane as

$$\begin{aligned} \begin{aligned} {\widetilde{\Omega }}_n(x)&= 1+\Gamma \left( \frac{n}{2}\right) \sum _{k=1}^{\infty } \frac{1}{\Gamma (k+\frac{n}{2}) k!} \left( \frac{x}{2}\right) ^{2k} \\ {}&= {\left\{ \begin{array}{ll} \Gamma \left( \frac{n}{2}\right) \left( \frac{2}{x}\right) ^{(n-2)/2} I_{(n-2)/2}(x), \qquad x \in {\mathbb {C}} \setminus \{0\}\\ 1, \qquad x = 0, \end{array}\right. } \end{aligned} \end{aligned}$$

(6)

where $I_{(n-2)/2}$ is the modified Bessel function of the first kind of order $(n-2)/2$ (Olver et al. 2010, formula 10.25.2).

Proposition 1

Let $\Omega _n$ and ${\widetilde{\Omega }}_n$ be the n-Schoenberg and the n-Nussbaum kernels as defined in Eqs. (4) and (6). Then, it is true that, for every $x \in {\mathbb {C}}$,

$$\begin{aligned} {\widetilde{\Omega }}_n(x) = {\widetilde{\Omega }}_n(-x) = \Omega _n(\textsf{i} x) = \Omega _n(-\textsf{i} x) \end{aligned}$$

and

$$\begin{aligned} \Omega _n(x) = \Omega _n(-x) = {\widetilde{\Omega }}_n(\textsf{i} x) = {\widetilde{\Omega }}_n(-\textsf{i} x). \end{aligned}$$

Proof

The proof comes by noting that the power series expansion (4) of $\Omega _n$ calculated at $\pm \textsf{i}x$ yields the power series expansion (6) of ${\widetilde{\Omega }}_n$ calculated at x or at $-x$, and vice-versa with the power series expansions (6) and (4) calculated at $\pm \textsf{i}x$ and $\pm x$, respectively. $\square$

Proposition 2

Let $r \in (0,\infty ]$ and $\psi : [0,r) \rightarrow {\mathbb {R}}$ be continuous. Then, $\psi$ belongs to $\text {EC}_n(r)$ if and only if it can be uniquely written as

$$\begin{aligned} \psi (x) = \int _{0}^{\infty } {\widetilde{\Omega }}_n (xu) \textrm{d} \gamma _n(u), \qquad x \in [0,r), \end{aligned}$$

(7)

where $\gamma _n$ is the n-Nussbaum measure of $\psi$ as defined in (5).

Proof

We start by invoking the integral representation provided by Widder (1934) (if $n=1$) and Devinatz (1955) (if $n>1$) for a radial exponentially convex function:

$$\begin{aligned} \psi (\Vert \varvec{x}\Vert _n) = \int _{{\mathbb {R}}^n} \textrm{e}^{\langle \varvec{x}, \varvec{\eta } u \rangle _n } \textrm{d} F(\varvec{u}), \quad \varvec{x}\in {\mathbb {B}}_n(r), \end{aligned}$$

(8)

where $\varvec{u}= \varvec{\eta } u$ with $\varvec{\eta } \in {\mathbb {S}}^{n-1}$ (unit $(n-1)$-dimensional sphere) and $u= \Vert \varvec{u}\Vert _n$, and F is a unique nonnegative finite Radon measure on ${\mathbb {R}}^n$. Since $\psi$ is radially symmetric, so will be F. Hence, we can factor F according to $\textrm{d} F (\varvec{u}) = \textrm{d} \chi _n(\varvec{\eta }) \textrm{d }\gamma _n(u)$, where $\chi _n$ is the Lebesgue measure on ${\mathbb {S}}^{n-1}$ and $\gamma _n$ is a unique nonnegative and finite Radon measure on $[0,\infty )$. Hence, the above integral can be rewritten as

$$\begin{aligned} \int _{[0,\infty ) } \Bigg ( \int _{{\mathbb {S}}^{n-1}} \textrm{e}^{\langle \varvec{x},\varvec{\eta } \rangle _n u} \textrm{d} \chi _n(\varvec{\eta }) \Bigg ) \textrm{d} \gamma _n(u). \end{aligned}$$

Setting $x=\Vert \varvec{x}\Vert _n$, we notice that the inner integral can be written in terms of polar coordinates as

$$\begin{aligned} \frac{\int _{0}^{\pi } \textrm{e}^{xu \cos \theta }(\sin \theta )^{n-2} \textrm{d} \theta }{ \int _{0}^{\pi } (\sin \theta )^{n-2} \textrm{d} \theta }, \end{aligned}$$

and formulae 3.621.1, 3.915.4 and 8.335.1 in Gradshteyn and Ryzhik (2014) lead to (7). The connection formulae in Proposition 1 prove that the Radon measures in (5) and (7) are the same, which establishes the uniqueness of the n-Nussbaum measure of $\psi$ and completes the proof. $\square$

Corollary 3

Let $\psi$ belong to $\text {EC}_n(\infty )$. Then, $x \mapsto \psi (\textsf{i} x)$ belongs to $\Phi _n(\infty )$.

Proof

This is a direct consequence of Eq. (3), Propositions 1 and 2, the n-Schoenberg measure of $x \mapsto \psi (\textsf{i} x)$ being the n-Nussbaum measure of $\psi$. $\square$

Corollary 3 provides a straightforward proof of Assertion a in Theorem 1 of Ehm et al. (2003) for the special case of radial exponential convexity. It might tempt us to believe that there exists a bijection between the classes $\Phi _n(\infty )$ and $\text {EC}_n(\infty )$. However, this was disproved in a counterexample by Ehm et al. (2003). In fact, it is needed that a radial positive definite function, $\varphi$, is additionally entire so that the function $x \mapsto \psi (x) = \varphi (- \textsf{i}x)$ is a radial exponentially convex function (Ehm et al. 2003, Theorem 1). Yet, this observation opens for some interesting work, that is summarized below.

Lemma 4

Let $0< r < \infty$ and $\varphi : [0,r)$ be an entire function that belongs to $\Phi _{n}(r)$. Then, $\varphi$ has an extension on $[0,\infty )$ that is entire and belongs to $\Phi _{n}(\infty )$, and this extension is unique.

Proof

We invoke Rudin’s extension theorem (Rudin 1970) to claim that $\varphi$ can be extended into a function ${\widetilde{\varphi }}:[0,\infty )$ that belongs to $\Phi _{n}(\infty )$. Because $\varphi$ is entire, ${\widetilde{\varphi }}$ is infinitely differentiable at the origin and its Taylor series at the origin has positive convergence radius. Furthermore, since ${\widetilde{\varphi }}$ is the radial part of a positive definite function, the infinite differentiability at the origin is equivalent to the infinite differentiability everywhere. This can be established recursively, by recalling that a stationary and isotropic random field is differentiable in the mean square sense if the radial part of its covariance function is differentiable at the origin, in which case it is differentiable everywhere and its second derivative is equal, up to the sign, to the radial part of the covariance function of the derivative random field, see Yaglom (1987); a covariance with a radial part that is infinitely differentiable at the origin is therefore associated with an infinitely differentiable random field and such a radial part is infinitely differentiable everywhere. The previous statements mean that ${\widetilde{\varphi }}$ is an analytic function that coincides with $\varphi$ on [0, r). The fact that ${\widetilde{\varphi }}$ is entire and unique then follows from Theorem 3.3.6.ii of Sasvári (2013). $\square$

Theorem 5

Let $\psi :[0,r) \rightarrow {\mathbb {R}}$ belong to $\text {EC}_n(r)$ with r a positive real number. Then, there exists a unique mapping ${\widetilde{\psi }}: [0,\infty ) \rightarrow {\mathbb {R}}$ that belongs to $\text {EC}_n(\infty )$, such that $\psi$ and ${\widetilde{\psi }}$ are identical on [0, r).

Proof

Arguments in Ehm et al. (2003, Theorem 1) imply that $\psi$ is entire and that the function $\varphi : x \mapsto \psi (\textsf{i}x)$ belongs to the class $\Phi _n(r)$. We invoke Lemma 4 to claim that there exists a unique entire mapping ${\widetilde{\varphi }}: [0,\infty ) \rightarrow {\mathbb {R}}$ belonging to $\Phi _n(\infty )$ such that $\varphi (x)={\widetilde{\varphi }}(x)$, for $x \in [0,r)$. Hence, we can use part b of Theorem 1 in Ehm et al. (2003) to claim that the mapping $x \mapsto {\widetilde{\psi }}(x):= {\widetilde{\varphi }}(-\textsf{i}x)$ belongs to $\text {EC}(\infty )$. The proof is completed by noting that ${\widetilde{\psi }}$ is identical to $\psi$ on [0, r). $\square$

Corollary 6

Let $\psi$ belong to $\text {EC}_n(r)$ with $r \in (0,\infty ]$. Then, $x \mapsto \psi (\textsf{i} x)$ belongs to $\Phi _n(r)$.

Proof

According to Theorem 5, the mapping $\psi$ has a unique extension to a radial exponentially convex function ${\widetilde{\psi }}$ on $[0,\infty )$. We now invoke Corollary 3, so that $x \mapsto {\widetilde{\psi }}(\textsf{i} x)$ belongs to $\Phi _n(\infty )$. In particular, its restriction to [0, r), which is the mapping $x \mapsto {\psi (\textsf{i} x)}$, belongs to $\Phi _n(r)$. The proof is complete. $\square$

3.2 The class $\text {EC}_n(\infty )$

Our research now focuses on the case $r=\infty$. Specifically, we are interested in a spectral inversion for the n-Nussbaum measures of members of the class $\text {EC}_n(\infty )$, for $n=1,2,\ldots$.

The motivation to look for a spectral inversion comes from a theorem in Silverman (1957), where it is stated that a function C in ${\mathbb {R}}\times {\mathbb {R}}$ as in (1) is a locally stationary covariance if and only if the function

$$\begin{aligned} \Psi (\omega ,\omega '):= \int \int \textrm{e}^{- \textsf{i} (\omega t - \omega ' t')} C(t,t') \textrm{d }t \textrm{d } t', \qquad \omega , \omega ' \in {\mathbb {R}}, \end{aligned}$$

is a locally stationary covariance function. His conclusion is unfortunately odd: the radial exponentially convex function in his Eq. 18 cannot be absolutely integrable with respect to the Lebesgue measure. Indeed, our Proposition 2 in concert with the properties of the n-Nussbaum kernel ${\widetilde{\Omega }}_n$ show that, unless it is identically equal to zero, a radial exponentially convex function in $\Phi _n(\infty )$ is never absolutely integrable with respect to the Lebesgue measure, as it is nonnegative and nondecreasing. A related remark is contained at page 2 of Silverman (1959), with an example at Eq. 3 in the same page, that artificially circumvents exponential convexity. The same remarks apply to Lemma 3 in Silverman (1959), for which special care is needed to ensure Fourier inversion.

Below, we prove that, under some mild regularity condition, the n-Nussbaum measure is absolutely continuous with respect to the Lebesgue measure, and we provide an explicit expression for its Radon-Nikodym derivative. Our conditions do not require pre-multiplying a given Gaussian random field by a Gaussian kernel to ensure integrability. Further, our result is valid for every integer n, while Silverman (1957, 1959) works for the case $n=1$ only.

Proposition 7

Let $\psi \in \text {EC}_n(\infty )$, such that $x \mapsto \psi (\textsf{i}x)$ is the radial part of an absolutely integrable function in ${\mathbb {R}}^n$. Then, the n-Nussbaum measure $\gamma _n$ of $\psi$ is absolutely continuous with respect to the Lebesgue measure and one has:

$$\begin{aligned} \frac{\textrm{d}\gamma _n(u)}{\textrm{d}u} = u^{n/2} \frac{2^{1-n/2}}{\Gamma \left( \frac{n}{2}\right) } \int _0^{\infty } x^{n/2} J_{(n-2)/2}(xu) \psi (\textsf{i}x) \textrm{d}x, \quad u \in [0,\infty ), \end{aligned}$$

(9)

with $u \mapsto u^{1-n} \frac{\textrm{d}\gamma _n(u)}{\textrm{d}u}$ being continuous and bounded on $[0,\infty )$.

Proof

On the one hand, as the mapping $x \mapsto \psi (\textsf{i}x)$ is the radial part of a positive definite function (Corollary 3) that is absolutely integrable in ${\mathbb {R}}^n$, it is the Hankel transform of order n of a bounded continuous function $f_n$ (Chilès and Delfiner 2012, formula 2.23):

$$\begin{aligned} \psi (\textsf{i}x) = (2\pi )^{n/2} x^{1-n/2} \int _0^{\infty } u^{n/2} J_{(n-2)/2}(xu) f_n(u) \textrm{d}u, \quad x \in [0,\infty ). \end{aligned}$$

(10)

On the other hand, Propositions 1 and 2 in concert with Eq. (4) show that

$$\begin{aligned} \begin{aligned} \psi (\textsf{i}x)&= \int _0^{\infty } \Omega _n(xu) \textrm{d}\gamma _n(u) \\&= \Gamma \left( \frac{n}{2}\right) 2^{n/2-1} x^{1-n/2} \int _0^{\infty } u^{1-n/2} J_{(n-2)/2}(xu) \textrm{d}\gamma _n(u), \quad x \in [0,\infty ). \end{aligned} \end{aligned}$$

(11)

Proposition 2 furthermore proves the uniqueness of the n-Nussbaum measure. Hence, (10) and (11) imply that $f_n(u)\textrm{d}u=\Gamma \left( \frac{n}{2}\right) 2^{-1}\pi ^{-n/2} u^{1-n}\textrm{d}\gamma _n(u)$, which entails that $u \mapsto u^{1-n} \frac{\textrm{d}\gamma _n(u)}{\textrm{d}u}$ is continuous and bounded on $[0,\infty )$. Finally, inverting the Hankel transform (Chilès and Delfiner 2012, formula 2.23) leads to (9). $\square$

Proposition 7 is the starting point to dig into the algebraic structure of n-Nussbaum measures. The result below shows that 1- and n-Nussbaum measures are recursively related through a linear operator that is reminiscent to the celebrated montée operator introduced by Matheron (1965).

Proposition 8

Let $\psi \in \text {EC}_n(\infty )$ such that $x \mapsto \psi (\textsf{i}x)$ is the radial part of an absolutely integrable function in ${\mathbb {R}}^n$. Then, $\psi$ has a n-Nussbaum measure $\gamma _n$ and a 1-Nussbaum measure $\gamma _1$ that are absolutely continuous with respect to the Lebesgue measure. Furthermore,

$$\begin{aligned} \frac{\textrm{d}\gamma _1(u)}{\textrm{d}u} = \frac{2\Gamma (\frac{n}{2})}{\sqrt{\pi }\Gamma (\frac{n-1}{2})} \int _u^{\infty } \left( v^2-u^2\right) ^{(n-3)/2} v^{2-n} \textrm{d}\gamma _n(v), \quad u \in [0,\infty ). \end{aligned}$$

(12)

Proof

As $\psi$ belongs to $\text {EC}_n(\infty )$, it also belongs to $\text {EC}_1(\infty )$, so it possesses a n-Nussbaum measure $\gamma _n$ and a 1-Nussbaum measure $\gamma _1$. The mapping $x \mapsto \psi (\textsf{i}x)$ is the radial part of an absolutely integrable covariance (Corollary 3) in ${\mathbb {R}}^n$, thus is it also the radial part of an absolutely integrable covariance in ${\mathbb {R}}$, which implies the existence of a n-dimensional spectral density $f_n$ and a 1-dimensional spectral density $f_1$ that are the Hankel transforms of order n and of order 1 of $x \mapsto \psi (\textsf{i}x)$:

$$\begin{aligned} f_n(u) = \frac{1}{(2\pi )^{n/2}} u^{1-n/2} \int _0^{\infty } x^{n/2} J_{n/2-1}(u x) \psi (\textsf{i}x) \textrm{d}x, \quad u \in [0,\infty ) \end{aligned}$$

(13)

and

$$\begin{aligned} f_1(u) = \frac{1}{(2\pi )^{1/2}} u^{1/2} \int _0^{\infty } x^{1/2} J_{-1/2}(u x) \psi (\textsf{i}x) \textrm{d}x, \quad u \in [0,\infty ), \end{aligned}$$

(14)

with (Chilès and Delfiner 2012, formula 2.29)

$$\begin{aligned} f_1(u) = \frac{2\pi ^{(n-1)/2}}{\Gamma (\frac{n-1}{2})} \int _u^{\infty } \left( v^2-u^2\right) ^{(n-3)/2} v f_n(v) \textrm{d}v, \quad u \in [0,\infty ). \end{aligned}$$

(15)

Comparing (13) and (14) with (9) yields the following identities:

$$\begin{aligned} \frac{\textrm{d}\gamma _n(u)}{\textrm{d}u} = \frac{2(\pi )^{n/2}}{\Gamma (\frac{n}{2})} u^{n-1} f_n(u), \quad u \in [0,\infty ) \end{aligned}$$

(16)

and

$$\begin{aligned} \frac{\textrm{d}\gamma _1(u)}{\textrm{d}u} = 2 f_1(u), \quad u \in [0,\infty ). \end{aligned}$$

(17)

Plugging (16) and (17) into (15) yields the claim. $\square$

Corollary 9

Under the conditions of Proposition 8, the n-Nussbaum measure is such that

$$\begin{aligned} \int _0^{\infty } v^{-1} \textrm{d}\gamma _n(v) < \infty . \end{aligned}$$

Proof

This is a consequence of the fact that the density $\frac{\textrm{d}\gamma _1(u)}{\textrm{d}u}$ is bounded (Proposition 7), in particular at $u=0$. $\square$

3.3 Examples

Examples of exponentially convex functions with a known analytical expression are scarce, see Silverman (1957) and Ehm et al. (2003). This implies that the available choice of locally stationary covariance functions as in (1) is extremely poor for the moment. Table 1 below provides a rich catalogue of radial exponentially convex functions over ${\mathbb {B}}_n(r)$, for $r\in (0,\infty ]$ and n a positive integer, together with their n-Nussbaum measures. Except for the first example, all the n-Nussbaum measures are absolutely continuous with respect to the Lebesgue measure; the analytic expressions can be obtained from the integral representation (7) by using formula 7.14.27 of Erdélyi (1953), formulae 19.5.3, 19.5.5, 19.5.6 and 19.5.7 of Erdélyi (1954), formulae 1.11.3.3, 2.15.2.3, 2.15.2.13, 2.15.10.3, 2.15.10.7, 2.15.14.1, 2.15.15.1 and 2.15.26.1 of Prudnikov et al. (1986), and formulae 10.43.23 and 10.43.28 of Olver et al. (2010).

Table 1 Examples of radial exponentially convex functions over ${\mathbb {B}}_n(r)$ (first column from left) and their associated radial positive definite functions (second column from left). In the third column from left, we provide the associated n-Nussbaum measures. B stands for the beta function, $\delta _{u=\alpha }$ for the Dirac measure centred on $\alpha$, ${\textsf {1}}_{u\in A}$ for the indicator of set A, equal to 1 if $u\in A$ and 0 otherwise, ${}_1F_1$ and ${}_1F_2$ for generalized hypergeometric functions (Olver et al. 2010, formula 16.2.1), erf and erfi for the real and imaginary error functions (Olver et al. 2010, formula 7.2.1), and $j_\beta$ for the first positive zero of $J_\beta$

3.4 Dimension walks on the class $\text {EC}_n(r)$

The fact that the classes $\text {EC}_n(r)$ for $n=1,2,\ldots$ are nested suggests that there might be projection operators relating elements of $\text {EC}_n(r)$ with those of $\text {EC}_{n'}(r)$, for given positive integers $n,n'$. The integral representation in terms of n-Nussbaum kernels and measures as in (7) suggests that such a search should start from the properties of the n-Nussbaum kernel, ${\widetilde{\Omega }}_n$.

Lemma 10

Let n be an integer greater than 2, and ${\widetilde{\Omega }}_n$ as defined in Eq. (6). Then, it is true that, for every $x > 0$,

$$\begin{aligned} \frac{n-2}{x} \frac{\textrm{d}{\widetilde{\Omega }}_{n-2}(x)}{\textrm{d}x} = {\widetilde{\Omega }}_{n}(x). \end{aligned}$$

Proof

The identity can be derived by expanding ${\widetilde{\Omega }}_{n-2}$ into the power series given in (6) and using a term-by-term differentiation. $\square$

Definition 11

Let n be an integer greater than 2 and $r \in (0,\infty ]$. Let $\psi$ belong to $\text {EC}_n(r)$ and $\gamma _n$ be its n-Nussbaum measure. If $\int _0^{\infty } {u^{-2}} \textrm{d}\gamma _n(u)<\infty$, the montée of order 2 of $\psi$ is the function defined on [0, r) by

$$\begin{aligned} {{\mathcal {M}}}_2\psi (x) = {2\pi (n-2)}\int _0^{\infty } {u^{-2}} \textrm{d}\gamma _n(u) + 2\pi \int _0^{x} v \psi (v) \textrm{d}v, \quad x \in [0,r). \end{aligned}$$

(18)

Since $\psi$ is an entire function, $v \mapsto v \, \psi (v)$ is integrable on any closed interval [0, x], so that the montée is always well defined.

Proposition 12

Let n be an integer greater than 2 and $r \in (0,\infty ]$. Let $\psi$ belong to $\text {EC}_n(r)$, such that the conditions of Definition 11 are met. Then $\mathcal{M}_2\psi$ belongs to $\text {EC}_{n-2}(r)$.

Proof

From the representation (7) of $\psi$ and Definition 11, one has

$$\begin{aligned} {{\mathcal {M}}}_2\psi (x) = {2\pi (n-2)}\int _0^{\infty } {u^{-2}} \textrm{d}\gamma _n(u) + 2\pi \int _0^{x} v \int _{0}^{\infty } {\widetilde{\Omega }}_{n} (vu) \, \textrm{d} \gamma _n(u) \textrm{d}v, \quad x \in [0,r).\nonumber \\ \end{aligned}$$

(19)

Using Fubini’s theorem and Lemma 10, this becomes

$$\begin{aligned} \begin{aligned} {{\mathcal {M}}}_2\psi (x)&= {2\pi (n-2)}\int _0^{\infty } {u^{-2}} \textrm{d}\gamma _n(u) + 2\pi \int _0^{\infty } \int _{0}^{x} v {\widetilde{\Omega }}_{n} (vu) \, \textrm{d}v \, \textrm{d} \gamma _n(u) \\&= {2\pi (n-2)}\int _0^{\infty } {u^{-2}} \textrm{d}\gamma _n(u) + 2\pi \int _0^{\infty } \frac{(n-2)}{u^2} \left[ {\widetilde{\Omega }}_{n-2} (xu)-{\widetilde{\Omega }}_{n-2} (0) \right] \textrm{d} \gamma _n(u) \\&= \int _0^{\infty } {\widetilde{\Omega }}_{n-2} (xu) \textrm{d} \mu _n(u), \quad x \in [0,r), \end{aligned} \end{aligned}$$

(20)

with $\textrm{d}\mu _n(u) = \frac{2\pi (n-2)}{u^2} \textrm{d}\gamma _n$ for $u \in (0,\infty )$. As $\gamma _n$ is a nonnegative Radon measure on $(0,\infty )$, so is $\mu _n$. Also, by hypothesis, $\mu _n$ is finite, which implies that ${{\mathcal {M}}}_2\psi$ belongs to $\text {EC}_{n-2}(r)$ based on Proposition 2. $\square$

3.5 Exponential convexity on product spaces

Let $n,n'$ be positive integers and $r,r'$ be positive real numbers. We define the class of componentwise radial exponentially convex functions over ${\mathbb {B}}_n(r) \times {\mathbb {B}}_{n'}(r')$ and denote $\text {EC}_{n,n'}(r,r')$ the class of continuous mappings $\psi : [0,r) \times [0,r') \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \sum _{i=1}^N \sum _{j=1}^N a_i {\overline{a}}_j \psi \left( \Vert \varvec{x}_i + \varvec{x}_j \Vert _n, \Vert \varvec{y}_i + \varvec{y}_j \Vert _{n'} \right) \ge 0, \end{aligned}$$

for every $\{a_i \}_{i=1}^N \subset {\mathbb {C}}$, $\{ \varvec{x}_i \}_{i=1}^N \subset {\mathbb {B}}_n(r/2)$ and $\{ \varvec{y}_i \}_{i=1}^N \subset {\mathbb {B}}_{n'}(r'/2)$.

Using similar techniques as in the previous section, it can be shown that $\psi \in \text {EC}_{n,n'}(r,r')$ if and only if

$$\begin{aligned} \psi (x,x') = \int _0^{\infty }\int _0^{\infty } {\widetilde{\Omega }}_n(xu) {\widetilde{\Omega }}_{n'}(x'u') \textrm{d} G_{n,n'}(u,u'), \qquad x \in [0,r), x' \in [0,r'), \end{aligned}$$

(21)

where $G_{n,n'}$ is a nonnegative and finite Radon measure, that we term $(n,n')$-Nussbaum measure, such the integral converges for every pair $(x,x')$.

Likewise, we write $\Phi _{n,n'}(r,r')$ for the class of componentwise radial positive definite functions over the product space ${\mathbb {B}}_n(r) \times {\mathbb {B}}_{n'}(r')$: $\varphi : [0,r) \times [0,r') \rightarrow {\mathbb {R}}$ belongs to $\Phi _{n,n'}(r,r')$ if

$$\begin{aligned} \sum _{i=1}^N \sum _{j=1}^N a_i {\overline{a}}_j \varphi \left( \Vert \varvec{x}_i - \varvec{x}_j \Vert _{n}, \Vert \varvec{y}_i - \varvec{y}_j \Vert _{n'} \right) \ge 0, \end{aligned}$$

for every $\{a_i \}_{i=1}^N \subset {\mathbb {C}}$, $\{ \varvec{x}_i \}_{i=1}^N \subset {\mathbb {B}}_n(r/2)$ and $\{ \varvec{y}_i \}_{i=1}^N \subset {\mathbb {B}}_{n'}(r'/2)$. Arguments in Porcu et al. (2006) show that $\varphi \in \Phi _{n,n'}(\infty ,\infty )$ if and only if

$$\begin{aligned} \varphi (x,x') = \int _0^{\infty }\int _0^{\infty } {\Omega }_n(xu) {\Omega }_{n'}(x'u') \textrm{d} F_{n,n'}(u,u'), \qquad x \in [0,\infty ), x' \in [0,\infty ), \end{aligned}$$

(22)

where $F_{n,n'}$ is a nonnegative and finite measure.

To the best of our knowledge, integral representations for the class $\Phi _{n,n'}(r,r')$ for finite $r,r'$ are not available. However, the following developments provide an extension theorem for members of the class $\text {EC}_{n,n'}(\infty ,r')$, as well as two examples of nonseparable functions belonging to $\text {EC}_{n,n'}(\infty ,r')$ and $\text {EC}_{n,n}(\infty ,\infty )$, respectively.

Lemma 13

Let $0< r' < \infty$ and $\varphi : [0,\infty ) \times [0,r')$ be an entire function that belongs to $\Phi _{n,n'}(\infty ,r')$ and such that $\varphi (\Vert \cdot \Vert _{n},x')$ is absolutely integrable in ${\mathbb {R}}^n$ for every $x'\in [0,r')$. Then, $\varphi$ has an extension on $[0,\infty ) \times [0,\infty )$ that is entire and belongs to $\Phi _{n,n'}(\infty ,\infty )$, and this extension is unique.

Proof

We invoke Theorem 3.2 in Porcu et al. (2023) to claim that $\varphi$ can be extended to a function ${\widetilde{\varphi }}:[0,\infty ) \times [0,\infty )$ that belongs to $\Phi _{n,n'}(\infty ,\infty )$. Following the statements in the proof of Lemma 4, it is seen that ${\widetilde{\varphi }}$ is the unique entire extension of $\varphi$. $\square$

Theorem 14

Let $0< r' < \infty$ and $\psi$ belong to $\text {EC}_{n,n'}(\infty ,r')$ such that $\psi (\textsf{i} \Vert \cdot \Vert _{n},\textsf{i} x')$ is absolutely integrable in ${\mathbb {R}}^n$ for every $x'\in [0,r')$. Then, $\psi$ admits a unique extension ${\widetilde{\psi }}$ that belongs to $\text {EC}_{n,n'}(\infty ,\infty )$.

Proof

Let $\psi \in \text {EC}_{n,n'}(\infty ,r')$. A straightforward generalization of Corollary 6 yields that the mapping $\varphi : (x,x') \mapsto \psi (\textsf{i}x, \textsf{i}x')$ belongs to the class $\Phi _{n,n'}(\infty ,r')$. Furthermore, $\varphi$ is entire, because so is $\psi$. We invoke Lemma 13 to claim that $\varphi$ admits a unique entire extension ${\widetilde{\varphi }}$ that belongs to $\Phi _{n,n'}(\infty ,\infty )$. Accordingly, based on Theorem 1 of Ehm et al. (2003), the mapping ${\widetilde{\psi }}: (x,x') \mapsto {\widetilde{\varphi }}(-\textsf{i}x,-\textsf{i}x')$ belongs to $\text {EC}_{n,n'}(\infty ,\infty )$. The proof is completed by noting that ${\widetilde{\psi }}$ is an extension of $\psi$ to $[0,\infty ) \times [0,\infty )$. $\square$

The current literature has no examples of parametric classes of componentwise exponentially convex functions defined over product spaces. Such examples are indeed important to obtain: for a given n and for $n'=1$, the construction of members of the class $\Phi _{n,n'}(r,r')$ is crucial to build classes of space-time locally stationary covariance functions according to (1). Below we present two examples of this kind, which do not belong to the separable class mentioned in Sect. 1.1.

Example 15

Consider the following density of a nonnegative Radon measure in $[0,\infty ) \times [0,\infty )$:

$$\begin{aligned} \frac{\textrm{d}G_{n,n'}(u,u')}{\textrm{d}u \, \textrm{d}u'}= {\left\{ \begin{array}{ll} \frac{4 u^{n-n'-1} u'^{n'-1}}{\Gamma \left( \frac{n}{2}\right) \Gamma \left( \frac{n'}{2}\right) } \exp \left( -\frac{\alpha u'^2}{u^2}-\beta u^2\right) , \quad u \in (0,\infty ), u' \in [0,\infty ),\\ 0, \qquad \qquad \qquad \qquad \qquad \qquad \qquad u = 0, u' \in [0,\infty ), \end{array}\right. } \end{aligned}$$

(23)

with $\alpha >0$ and $\beta >0$. Then, using the 13-th entry of Table 1, one gets

$$\begin{aligned} \int _0^{\infty } \Omega _n(x' u') \frac{\textrm{d}G_{n,n'}(u,u')}{\textrm{d}u \, \textrm{d}u'} \textrm{d}u' = \frac{2\alpha ^{-\frac{n'}{2}}}{\Gamma \left( \frac{n}{2}\right) } u^{n-1} \exp \left( -\left( \beta -\frac{x'^2}{4\alpha }\right) u^2\right) , \quad u,x' \in [0,\infty ),\nonumber \\ \end{aligned}$$

(24)

and, if $x' \le 2 \sqrt{\alpha \beta }$,

$$\begin{aligned} \begin{aligned} \psi (x,x')&:= \int _0^{\infty } \int _0^{\infty } \Omega _n(x u) \Omega _n(x' u') \textrm{d}G_{n,n'}(u,u') \\&= 2^n \alpha ^{\frac{n-n'}{2}} \left( 4\alpha \beta -x'^2\right) ^{-\frac{n}{2}} \exp \left( \frac{\alpha x^2}{4\alpha \beta -x'^2}\right) , \quad x \in [0,\infty ), x' \in [0,2 \sqrt{\alpha \beta }). \end{aligned} \end{aligned}$$

(25)

One therefore obtains a componentwise radial and nonseparable exponentially convex function over ${\mathbb {R}}^n \times {\mathbb {B}}_{n'}(2 \sqrt{\alpha \beta })$, associated with the $(n,n')$-Nussbaum measure $G_{n,n'}$ defined in (23). The associated radial positive definite function is

$$\begin{aligned} \varphi (x,x') = 2^n \alpha ^{\frac{n-n'}{2}} \left( 4\alpha \beta +x'^2\right) ^{-\frac{n}{2}} \exp \left( -\frac{\alpha x^2}{4\alpha \beta +x'^2}\right) , \quad x \in [0,\infty ), x' \in [0,2 \sqrt{\alpha \beta }),\nonumber \\ \end{aligned}$$

(26)

which belongs to the Gneiting class of nonseparable positive definite functions (Gneiting 2002b).

Example 16

A similar construction with the 16-th entry of Table 1 leads to the following (n, n)-Nussbaum density, where $\alpha$, $\beta$ and $\gamma$ are positive real numbers such that $4\beta \gamma > \alpha ^2$:

$$\begin{aligned} \frac{\textrm{d}G_{n,n}(u,u')}{\textrm{d}u \textrm{d}u'} = (4 \beta \gamma -\alpha ^2) (uu')^{\frac{n}{2}} \exp (-\beta u^2 -\gamma u'^2) I_{\frac{n}{2}-1}(\alpha u u'), \quad u,u' \in [0,\infty ),\nonumber \\ \end{aligned}$$

(27)

associated with the following nonseparable radial exponentially convex function over ${\mathbb {R}}^n \times {\mathbb {R}}^n$:

$$\begin{aligned} \psi (x,x') = \Gamma ^2\left( \frac{n}{2}\right) \left( \frac{4}{xx'}\right) ^{\frac{n}{2}-1} \exp \left( \frac{\gamma x^2+\beta x'^2}{4\beta \gamma -\alpha ^2}\right) I_{\frac{n}{2}-1}\left( \frac{\alpha x x'}{4\beta \gamma -\alpha ^2}\right) , \quad x,x'\in [0,\infty ),\nonumber \\ \end{aligned}$$

(28)

and with the following nonseparable radial positive definite function over ${\mathbb {R}}^n \times {\mathbb {R}}^n$:

$$\begin{aligned} \varphi (x,x') = \Gamma ^2\left( \frac{n}{2}\right) \left( \frac{4}{xx'}\right) ^{\frac{n}{2}-1} \exp \left( \frac{\gamma x^2+\beta x'^2}{\alpha ^2-4\beta \gamma }\right) J_{\frac{n}{2}-1}\left( \frac{\alpha x x'}{4\beta \gamma -\alpha ^2}\right) , \quad x,x'\in [0,\infty ),\nonumber \\ \end{aligned}$$

(29)

which does not belong to the Gneiting class as it can take negative values.

4 Concluding remarks

We have provided a collection of results that enrich the knowledge of exponentially convex functions, for which the literature is scarce. Further than that, we have given several examples of radial exponentially convex functions that can be used to build new classes of locally stationary covariance functions in the sense of Silverman (1957). We note that, up to now, only a couple of examples of locally stationary covariance functions have been available in the literature. Our catalogue provides a wealth of examples by combining the entries in Table 1 or the examples in Sect. 3.5 with any of the stationary covariance functions available in the geostatistical literature, see Chilès and Delfiner (2012) and the reviews in Porcu et al. (2018) and Porcu et al. (2021). Our results might turn very useful to provide reduction techniques for locally stationary processes into stationary ones, as in Perrin and Senoussi (2000) and more recently in Porcu et al. (2020).