On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives

Abstract

We use the theory of Bernstein functions to analyze power law tail behavior with log-periodic perturbations which corresponds to self-similarity of the Bernstein functions. Such tail behavior appears in the context of semistable Lévy processes. The Bernstein approach enables us to solve some open questions concerning semi-fractional derivatives recently introduced in Kern et al. (Fract Calc Appl Anal 22(2):326–357, 2019) by means of the generators of certain semistable Lévy processes. In particular, it is shown that semi-fractional derivatives can be seen as generalized fractional derivatives in the sense of Kochubei (Integr Equ Oper Theory 71:583–600, 2011) and generalized fractional derivatives can be constructed by means of arbitrary Bernstein functions vanishing at the origin.

1 Introduction

A non-negative function $\widetilde{\psi }:(0,\infty )\rightarrow (0,\infty )$ is called a Bernstein function if it is of class $C^{\infty }(0,\infty )$ and

$$\begin{aligned} (-1)^{n-1}\widetilde{\psi }^{(n)}(x)\ge 0\quad \text {for all}\; n\in {\mathbb N}\,\text {and}\, x>0. \end{aligned}$$

(1.1)

Its first derivative $f=\widetilde{\psi }'$ is a completely monotone function, i.e.

$$\begin{aligned} (-1)^{n}f^{(n)}(x)\ge 0\quad \text {for all}\; n\in {\mathbb N}_{0}\, \text {and}\, x>0. \end{aligned}$$

(1.2)

Due to a celebrated result of Bernstein, the completely monotone function f is the Laplace transform of a unique Borel measure $\mu$ on $[0,\infty )$

$$\begin{aligned} f(x)=\widetilde{\mu }(x):=\int _{0}^{\infty }e^{-xt}\,\hbox {d}\mu (t). \end{aligned}$$

(1.3)

As a consequence, the Bernstein function admits the representation

$$\begin{aligned} \widetilde{\psi }(x)=a+bx+\int _{0}^{\infty }\left( 1-e^{-xt}\right) \,\hbox {d}\phi (t) \end{aligned}$$

(1.4)

for a unique triplet $[a,b,\phi ]$, where $a,b\ge 0$ and $\phi$ is a Borel measure on $(0,\infty )$ satisfying $\int _{0}^{\infty }\min \{1,t\}\,\hbox {d}\phi (t)<\infty$, also called the Lévy measure. For details on Bernstein functions, completely monotone functions and their connection to stochastic processes we refer to the monograph [31]. It is well known that in case $a=b=0$ the Bernstein function $\widetilde{\psi }(x)=\int _{0}^{\infty }\left( 1-e^{-xt}\right) \,\hbox {d}\phi (t)$ is the Laplace exponent of a Lévy subordinator $(X_{t})_{t\ge 0}$, i.e. ${\mathbb E}[\exp (-sX_{t})]=\exp (-t\cdot \widetilde{\psi }(s))$ for all $t\ge 0$, $s>0$, where $(X_{t})_{t\ge 0}$ is a Lévy process with almost surely non-decreasing sample paths.

In Sect. 2 we will introduce a self-similarity property for Bernstein functions which is intimately connected to the following class of functions.

Definition 1.1

We call a function $\theta :{\mathbb R}\rightarrow {\mathbb R}$ admissible with respect to the parameters $\alpha \in (0,2){\setminus }\{1\}$ and $c>1$ if the following three conditions are fulfilled:

$$\begin{aligned}&\theta (x)>0 \text { for all }x\in {\mathbb R}, \end{aligned}$$

(1.5)

$$\begin{aligned}&\text {the mapping } t\mapsto t^{-\alpha }\theta (\log t)\text { is non-increasing for }t>0, \end{aligned}$$

(1.6)

$$\begin{aligned}&\theta \text { is }\log (c^{1/\alpha })\text {-periodic.} \end{aligned}$$

(1.7)

We will frequently require that an admissible function $\theta$ is smooth, meaning that it is continuous and piecewise continuously differentiable.

In case $\alpha \in (0,1)$ we use an admissible function $\theta$ to define

$$\begin{aligned} \phi (t,\infty ):=t^{-\alpha }\theta (\log t)\quad \text { for all }\; t>0 \end{aligned}$$

(1.8)

as the positive tail of a Lévy measure $\phi$ concentrated on $(0,\infty )$ which belongs to a semistable distribution $\nu$ with log-characteristic function

$$\begin{aligned} \psi (x)=\int _{0}^\infty \left( e^{ixy}-1\right) \,\hbox {d}\phi (y)\quad \text { for all }\;x\in {\mathbb R}\end{aligned}$$

(1.9)

given uniquely by the Fourier transform $\widehat{\nu }(x)=\int _{\mathbb R}e^{ixy}\,d\nu (y)=\exp (\psi (x))$. The corresponding Lévy process $(X_{t})_{t\ge 0}$ given by ${\mathbb E}[\exp (ix\cdot X_{t})]=\exp (t\cdot \psi (x))$ for all $t\ge 0$ and $x\in {\mathbb R}$ is called a semistable subordinator. For details on semistable distributions and Lévy processes we refer to the monographs [21, 30]. The power law tail behavior with log-periodic perturbations of an admissible function $\theta$ naturally appears in various applications of natural sciences and other areas; see [34] and section 5.4 in [35]. In recent years the asymptotic fine structure of the corresponding measure $\phi$ and the function $\psi$ has drawn some attention; cf. [14, 15]. Given an admissible function $\theta$ with respect to $\alpha \in (0,1)$ and $c>1$ with corresponding semistable Lévy measure $\phi$ given by (1.8), by Definition 2.2 in [13] the semi-fractional derivative $\frac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }$ of order $\alpha \in (0,1)$ is given by the non-local operator

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }\,f(x):=-Lf(x)=\int _0^\infty \left( f(x)-f(x-y)\right) \,\hbox {d}\phi (y), \end{aligned}$$

(1.10)

where L is the generator of the corresponding semistable Lévy process and at least functions f in the Sobolev space $W^{2,1}({\mathbb R})$ belong to the domain of the semi-fractional derivative. In terms of the Fourier transform we can equivalently rewrite (1.10) as

$$\begin{aligned} \widehat{\tfrac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }\,f}(x):=-\widehat{Lf}(x)=-\psi (x)\,\widehat{f}(x)\quad \text { for all }x\in {\mathbb R}, \end{aligned}$$

(1.11)

where the Fourier transform of f is given by $\widehat{f}(x)=\int _{\mathbb R}e^{ixy}f(y)\,dy$; see [13] for details. In Sect. 2, we will start with the elementary observation that for $\alpha \in (0,1)$ there is a one-to-one correspondence between self-similar Bernstein functions given as the Laplace exponent $\widetilde{\psi }(x)=-\psi (ix)$ for $x>0$ and semistable Lévy measures $\phi$ of the form (1.8). In particular, this enables us to show that semi-fractional derivatives of order $\alpha \in (0,1)$ can be seen as a special case of generalized fractional derivatives in the sense of [17]. A constant function $\theta$ corresponds to the complete Bernstein function $\widetilde{\psi }(x)=x^{\alpha }$ and an ordinary fractional derivative of order $\alpha \in (0,1)$. For details on classical fractional derivatives we refer to the monographs [16, 26, 28]. In fact we will show that the construction of generalized fractional derivatives is not restricted to self-similar Bernstein functions and holds for arbitrary Bernstein functions with $a=b=0$ in (1.4).

In Sect. 3 we will prove a discrete approximation formula of the generator in (1.10) involving a generalized Sibuya distribution given in terms of the self-similar Bernstein function.

In case $\alpha \in (1,2)$ we use an admissible function $\theta$ to define

$$\begin{aligned} \phi (-\infty ,-t):=t^{-\alpha }\theta (\log t)\quad \text { for all }\;t>0 \end{aligned}$$

(1.12)

as the negative tail of a Lévy measure $\phi$ concentrated on $(-\infty ,0)$ which belongs to a different semistable distribution $\nu$ with log-characteristic function

$$\begin{aligned} \psi (x)=\int _{-\infty }^0\left( e^{ixy}-1-ixy\right) \,\hbox {d}\phi (y)\quad \text { for all }x\in {\mathbb R}. \end{aligned}$$

(1.13)

Given an admissible function $\theta$ with respect to $\alpha \in (1,2)$ and $c>1$ with corresponding semistable Lévy measure $\phi$ given by (1.12), according to Definition 2.5 in [13] the negative semi-fractional derivative $\frac{\partial ^\alpha }{\partial _{c,\theta }(-x)^\alpha }$ of order $\alpha \in (1,2)$ is given by the non-local operator

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }(-x)^\alpha }\,f(x):=Lf(x)=\int _0^\infty \left( f(x+y)-f(x)-y\,f'(x)\right) \,\hbox {d}\phi (-y), \end{aligned}$$

(1.14)

where L is again the generator of the corresponding semistable Lévy process and at least functions f in the Sobolev space $W^{2,1}({\mathbb R})$ belong to the domain of the semi-fractional derivative. For $\alpha \in (1,2)$ the function $x\mapsto \psi (-ix)$ cannot be a Bernstein function, but we will show in Sect. 4 that it has an inverse which is a self-similar Bernstein function. This will enable us to solve an open question from [12] concerning space-time duality for semi-fractional differential equations.

2 Self-Similar Bernstein Functions and Semi-Fractional Derivatives

In this section we suppose that $\alpha \in (0,1)$ in which case the Laplace exponent $\widetilde{\psi }$ of the corresponding semistable distribution $\nu$, uniquely given by the Laplace transform $\widetilde{\nu }(x)=\int _0^\infty e^{-xy}\,d\nu (y)=\exp (-\widetilde{\psi }(x))$ for $x>0$, in view of (1.9) can be represented as

$$\begin{aligned} \widetilde{\psi }(x) =-\psi (ix)=\int _0^\infty \left( 1-e^{-xy}\right) \,\hbox {d}\phi (y). \end{aligned}$$

(2.1)

Clearly, $\widetilde{\psi }$ is a Bernstein function since $\phi$ integrates $\min \{1,t\}$ on $(0,\infty )$.

Definition 2.1

We call a Bernstein function $\widetilde{\psi }$ self-similar with respect to $\alpha \in (0,1)$ and $c>1$ if it admits the discrete scale invariance

$$\begin{aligned} \widetilde{\psi }(c^{1/\alpha }x)=c\cdot \widetilde{\psi }(x)\quad \text { for all}\; x>0. \end{aligned}$$

Clearly, self-similarity can be iterated leading to $\widetilde{\psi }(c^{m/\alpha }x)=c^{m}\cdot \widetilde{\psi }(x)$ for all $x>0$ and $m\in {\mathbb Z}$. Moreover, the set of self-similarity factors

$$\begin{aligned} {\mathbb {S}}=\left\{ c>0: \widetilde{\psi }(c^{1/\alpha }x)=c\cdot \widetilde{\psi }(x)\text { for all }x>0\right\} \end{aligned}$$

(2.2)

is a closed multiplicative subgroup of $(0,\infty )$ due to continuty of $\widetilde{\psi }$, such that we either have the discrete scale invariance ${\mathbb {S}}=c^{{\mathbb Z}}$ for some $c>1$, or continuous scaling ${\mathbb {S}}=(0,\infty )$. Our focus is on the discrete scale behavior and we will give an impression of how corresponding self-similar Bernstein functions look like after the following elementary observation, which is our key result.

Lemma 2.2

For fixed $\alpha \in (0,1)$ and $c>1$ the following statements are equivalent.

• (i) $\widetilde{\psi }$ is a self-similar Bernstein function with respect to $\alpha \in (0,1)$ and $c>1$.

• (ii) $\widetilde{\psi }(x)=\int _0^\infty \left( 1-e^{-xy}\right) \,\hbox {d}\phi (y)$, where the Lévy measure $\phi$ is given by (1.12) for some admissible function $\theta$ with respect to $\alpha \in (0,1)$ and $c>1$.

In either case we have $\widetilde{\psi }(x)=x^\alpha \gamma (-\log x)$, where

$$\begin{aligned} \gamma (x)=\int _{0}^{\infty }e^{-t}t^{-\alpha }\theta (\log t+x)\,\hbox {d}t \end{aligned}$$

(2.3)

is an admissible $C^\infty ({\mathbb R})$-function with respect to $\alpha \in (0,1)$ and $c>1$.

Proof

“(i)$\Rightarrow$(ii)”: Since $\widetilde{\psi }$ is a Bernstein function, by (1.4) we have

$$\begin{aligned} \widetilde{\psi }(x)=a+bx+\int _0^\infty (1-e^{-xt})\,\hbox {d}\phi (t) \end{aligned}$$

for some $a,b\ge 0$ and a Lévy measure $\phi$ on $(0,\infty )$ integrating $\min \{1,t\}$. Iterating the self-similarity relation shows that $\widetilde{\psi }(c^{m/\alpha }x)=c^m\widetilde{\psi }(x)$ for all $x>0$ and $m\in {\mathbb Z}$. As $m\rightarrow -\infty$, we see that $\lim _{x\downarrow 0}\widetilde{\psi }(x)=0$ and hence $a=0$. By the transformation rule self-similarity now reads as

$$\begin{aligned} \begin{aligned} \widetilde{\psi }(c^{1/\alpha }x)&=b\,c^{1/\alpha }x+\int _0^\infty (1-e^{-xt})\,\hbox {d}(c^{1/\alpha }\phi )(t)\\&=b\,cx+c\cdot \int _0^\infty (1-e^{-xt})\,\hbox {d}\phi (t)=c\cdot \widetilde{\psi }(x), \end{aligned} \end{aligned}$$

(2.4)

where $(c^{1/\alpha }\phi )$ denotes the image measure under scale multiplication with $c^{1/\alpha }$. Since the Bernstein triplet $[a,b,\phi ]$ is unique, we must have $b=0$ and $(c^{1/\alpha }\phi )=c\cdot \phi$. Hence by Lemma 7.1.6 and Corollary 7.4.4 in [21] we have that $\phi$ is a $(c^{1/\alpha },c)$-semistable Lévy measure on $(0,\infty )$ fulfilling (1.12) for some admissible function $\theta$ with respect to $\alpha \in (0,1)$ and $c>1$. The connection of Corollary 7.4.4 in [21] to admissible functions is made precise by Lemma A.1 in the Appendix.

“(ii)$\Rightarrow$(i)”: Clearly, $\widetilde{\psi }$ is a Bernstein function by Theorem 3.2 in [31] and self-similarity follows from (2.4) with $b=0$ which is valid by Lemma 7.1.6 in [21].

Now define $\gamma (x):=e^{\alpha x}\widetilde{\psi }(e^{-x})$ which is of class $C^\infty ({\mathbb R})$ by the product rule and $\gamma (x)>0$ for all $x\in {\mathbb R}$. Then $\gamma$ is $\log (c^{1/\alpha })$-periodic, since

$$\begin{aligned} \gamma (x+\log (c^{1/\alpha }))=e^{\alpha x}c\cdot \widetilde{\psi }(c^{-1/\alpha }e^{-x})=e^{\alpha x}\widetilde{\psi }(e^{-x})=\gamma (x), \end{aligned}$$

and $t\mapsto t^{-\alpha }\gamma (\log t)=\widetilde{\psi }(t^{-1})$ is non-increasing. Hence $\gamma$ is an admissible function with respect to $\alpha \in (0,1)$ and $c>1$. Moreover, by partial integration and a change of variables $t=xy$ we get

$$\begin{aligned} \widetilde{\psi }(x)&=\int _0^\infty \left( 1-e^{-xy}\right) \,\hbox {d}\phi (y)= x\int _0^\infty e^{-xy}y^{-\alpha }\theta (\log y)\,\hbox {d}y\\&=x^{\alpha }\int _0^\infty e^{-t}t^{-\alpha }\theta (\log t-\log x)\,\hbox {d}t=x^{\alpha }\gamma (-\log x), \end{aligned}$$

which shows that (2.3) holds, concluding the proof. $\square$

Example 2.3

Since $\gamma$ in (2.3) is admissible with respect to the same parameters $\alpha \in (0,1)$ and $c>1$ as $\theta$, we may iterate the procedure. Starting with $\gamma _{0}:=\theta$ and setting

$$\begin{aligned} \gamma _{k}(x):=\int _{0}^{\infty }e^{-t}t^{-\alpha }\gamma _{k-1}(\log t+x)\,\hbox {d}t \end{aligned}$$

for $k\in {\mathbb N}$, we have $\gamma _{1}=\gamma$ and the sequence $(\widetilde{\psi }_{k})_{k\in {\mathbb N}}$ given by $\widetilde{\psi }_{k}(x)=x^{\alpha }\gamma _{k}(-\log x)$ is a sequence of self-similar Bernstein functions by Lemma 2.2.

Example 2.4

If ${\mathbb {S}}=(0,\infty )$ in (2.2), then by Lemma 2.2 the function $\theta$ and hence also the function $\gamma$ must be constant. Thus $\widetilde{\psi }$ is the well known prototype of a Bernstein function $\widetilde{\psi }(x)=C\,x^{\alpha }$ for some $C>0$ and $\alpha \in (0,1)$. To give an explicit example with ${\mathbb {S}}=c^{{\mathbb Z}}$, for fixed $\alpha \in (0,1)$ we consider the simple periodic function $\theta (x)=1+\frac{1}{\alpha }+\sin x$, which is positive and $\log (c^{1/\alpha })$-periodic for $c=e^{2\pi \alpha }$. Moreover, $\theta$ is a smooth admissible function by Lemma A.1(v) in the Appendix, since $\theta '(x)=\cos x\le 1+\alpha (1+\sin x)=\alpha \theta (x)$. For the corresponding function $\gamma$ we have

$$\begin{aligned} \gamma (x)&=\int _{0}^{\infty }e^{-t}t^{-\alpha }\left( 1+\tfrac{1}{\alpha }+\sin (\log t+x)\right) \,\hbox {d}t\\&=(1+\tfrac{1}{\alpha })\,\Gamma (1-\alpha )+\int _{0}^{\infty }e^{-t}t^{-\alpha }\sin (\log t+x)\,\hbox {d}t \end{aligned}$$

and hence the self-similar Bernstein function has the representation

$$\begin{aligned} \widetilde{\psi }(x)=x^{\alpha }\left( (1+\tfrac{1}{\alpha })\,\Gamma (1-\alpha )+\int _{0}^{\infty }e^{-t}t^{-\alpha }\sin (\log t-\log x)\,\hbox {d}t\right) . \end{aligned}$$

Now, for $c=e^{2\pi \alpha }$ we have

$$\begin{aligned} \gamma \left( -\log \left( \sqrt{c}^{1/\alpha }x\right) \right)&=(1+\tfrac{1}{\alpha })\,\Gamma (1-\alpha )+\int _{0}^{\infty }e^{-t}t^{-\alpha }\sin (\log t-\log x-\pi )\,\hbox {d}t\\&=(1+\tfrac{1}{\alpha })\,\Gamma (1-\alpha )-\int _{0}^{\infty }e^{-t}t^{-\alpha }\sin (\log t-\log x)\,\hbox {d}t\\&=\gamma (-\log x)-2\int _{0}^{\infty }e^{-t}t^{-\alpha }\sin (\log t-\log x)\,\hbox {d}t, \end{aligned}$$

which cannot coincide with $\gamma (-\log x)$ for all $x>0$. Hence $\widetilde{\psi }(\sqrt{c}^{1/\alpha }x)\not \equiv \sqrt{c}\widetilde{\psi }(x)$ and we have ${\mathbb {S}}=c^{{\mathbb Z}}$. Iterating the procedure as in Example 2.3 gives us

$$\begin{aligned} \gamma _{k}(x)=(1+\tfrac{1}{\alpha })\,(\Gamma (1-\alpha ))^{k}+f_{k}(x) \end{aligned}$$

for some non-constant and periodic function $f_{k}$ with $|f_{k}(x)|\le (\Gamma (1-\alpha ))^{k}$ for all $x\in {\mathbb R}$ and $k\in {\mathbb N}$. Since $\Gamma (1-\alpha )>1$ this shows that in general we cannot expect convergence of the sequence $(\widetilde{\psi }_{k})_{k\in {\mathbb N}}$ in Example 2.3.

Remark 2.5

If we assume that the admissible function $\theta$ is smooth, then it admits a Fourier series representation

$$\begin{aligned} \theta (x)=\sum _{k\in {\mathbb Z}}c_k\,e^{ik\tilde{c}x}\quad \text { with }\quad \tilde{c}=\frac{2\pi \alpha }{\log c}. \end{aligned}$$

(2.5)

In this case the function $\gamma$ appearing in Lemma 2.2 is given by the modified Fourier series

$$\begin{aligned} \gamma (x)=\sum _{k\in {\mathbb Z}}c_k\,\Gamma (ik\tilde{c}-\alpha +1)\,e^{ik\tilde{c}x}\quad \text { for }x\in {\mathbb R}\end{aligned}$$

(2.6)

which can be seen as follows. By Theorem 3.1 in [13] the coefficients of this series appear in a representation of the log-characteristic function

$$\begin{aligned} \psi (x)=-\sum _{k\in {\mathbb Z}}c_k\,\Gamma (ik\tilde{c}-\alpha +1)\,(-ix)^{\alpha -ik\tilde{c}}\quad \text { for }x\in {\mathbb R}\end{aligned}$$

(2.7)

and the relation $\gamma (x)=e^{\alpha x}\,\widetilde{\psi }(e^{-x})=-e^{\alpha x}\psi (ie^{-x})$ easily shows (2.6).

A natural question which arises is, if for two admissible functions $\theta _1,\,\theta _2$ with respect to the parameters $\alpha _1\in (0,1)$ and $c_1>1$, respectively $\alpha _2\in (0,1)$ and $c_2>1$, the composition of the corresponding semi-fractional derivatives given by (1.10) can again be a semi-fractional derivative of order $\alpha :=\alpha _1+\alpha _2$. We concentrate on the easiest case when $\alpha \in (0,1)$ and $\theta _1,\,\theta _2$ have the same periodicity, i.e. $c_1^{1/\alpha _1}=c_2^{1/\alpha _2}$. In view of (1.11) for the corresponding log-characteristic functions we need to show that

$$\begin{aligned} \psi _1(x)\cdot \psi _2(x)=-\psi (x)\quad \text { for all }x\in {\mathbb R}, \end{aligned}$$

(2.8)

where $\psi$ is the log-characteristic function of a semistable distribution corresponding to an admissible function $\theta$ with respect to the parameters $\alpha \in (0,1)$ and $c:=c_1^{\alpha /\alpha _1}=c_2^{\alpha /\alpha _2} >1$. Since (2.8) is equivalent to $\widetilde{\psi }_1\cdot \widetilde{\psi }_2=\widetilde{\psi }$ for the corresponding log-Laplace exponents and we have self-similarity

$$\begin{aligned} \widetilde{\psi }(c^{1/\alpha }x)= & {} \widetilde{\psi }_1(c_1^{1/\alpha _1}x)\cdot \widetilde{\psi }_2(c_2^{1/\alpha _2}x)=c_1c_2\cdot \widetilde{\psi }_1(x)\widetilde{\psi }_2(x)\\= & {} c^{\alpha _1/\alpha }c^{\alpha _2/\alpha }\cdot \widetilde{\psi }(x)=c\cdot \widetilde{\psi }(x), \end{aligned}$$

in view of Lemma 2.2 this is equivalent to require that $\widetilde{\psi }$ is a Bernstein function.

Corollary 2.6

$\widetilde{\psi }_1\cdot \widetilde{\psi }_2$ is a Bernstein function iff there exists an admissible function $\theta$ with respect to the parameters $\alpha =\alpha _1+\alpha _2\in (0,1)$ and $c=c_1^{\alpha /\alpha _1}=c_2^{\alpha /\alpha _2}>1$ such that

$$\begin{aligned} \frac{\partial ^{\alpha _2}}{\partial _{c_2,\theta _2}x^{\alpha _2}}\,\frac{\partial ^{\alpha _1}}{\partial _{c_1,\theta _1}x^{\alpha _1}}\,f=\frac{\partial ^{\alpha }}{\partial _{c,\theta }x^{\alpha }}\,f \end{aligned}$$

(2.9)

for suitable functions $f\in W^{2,1}({\mathbb R})$ with $\frac{\partial ^{\alpha _1}}{\partial _{c_1,\theta _1}x^{\alpha _1}}\,f$ belonging to the domain of $\frac{\partial ^{\alpha _2}}{\partial _{c_2,\theta _2}x^{\alpha _2}}$.

Remark 2.7

In case $\theta _1,\,\theta _2$ are smooth admissible functions with Fourier series representations as in (2.5)

$$\begin{aligned} \theta _1(x)=\sum _{k\in {\mathbb Z}}c_{k,1}\,e^{ik\tilde{c}x}\quad \text { and }\quad \theta _2(x)=\sum _{\ell \in {\mathbb Z}}c_{\ell ,2}\,e^{i\ell \tilde{c}x}, \end{aligned}$$

where $\tilde{c}=\frac{2\pi \alpha _1}{\log c_1}=\frac{2\pi \alpha _2}{\log c_2}=\frac{2\pi \alpha }{\log c}$, then by (2.7) and the Cauchy product rule we easily get

$$\begin{aligned} \widetilde{\psi }(x)=\widetilde{\psi }_1(x)\cdot \widetilde{\psi }_2(x)=\sum _{m\in {\mathbb Z}}d_m\,\Gamma (im\tilde{c}-(\alpha _1+\alpha _2)+1)\,x^{\alpha _1+\alpha _2-im\tilde{c}}, \end{aligned}$$

(2.10)

where

$$\begin{aligned} d_m=\sum _{\ell \in {\mathbb Z}}c_{m-\ell ,1}c_{\ell ,2}\,\frac{\Gamma (i(m-\ell )\tilde{c}-\alpha _1+1)\,\Gamma (i\ell \tilde{c}-\alpha _2+1)}{\Gamma (im\tilde{c}-(\alpha _1+\alpha _2)+1)}. \end{aligned}$$

Uniqueness of the Fourier coefficients then gives us the representation

$$\begin{aligned} \theta (x)=\sum _{m\in {\mathbb Z}}d_m\,e^{im\tilde{c}x} \end{aligned}$$

(2.11)

for the admissible function $\theta$ from Corollary 2.6, provided that (2.10) is a Bernstein function.

Finally, we want to show that the semi-fractional derivative of order $\alpha \in (0,1)$ can be seen as a special case of a generalized fractional derivative introduced in [17]. Starting with (1.10) for a smooth admissible function $\theta$, integration by parts shows that

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }\,f(x)=\int _0^\infty f'(x-y)\,y^{-\alpha }\theta (\log y)\,\hbox {d}y \end{aligned}$$

as laid out in [13]. Introducing the kernel function $k(y):=y^{-\alpha }\theta (\log y)$ and restricting considerations to functions with support on the positive real line, this can be interpreted as a semi-fractional derivative of Caputo type

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }\,f(x)=\int _0^x f'(x-y)\,k(y)\,dy=(k*f')(x). \end{aligned}$$

On the other hand, interchanging the order of integration and differentiation gives a semi-fractional derivative of Riemann–Liouville type

$$\begin{aligned} {\mathbb {D}}_{c,\theta }^\alpha f(x)=\frac{\hbox {d}}{\hbox {d}x}\int _0^x f(x-y)\,k(y)\,\hbox {d}y=(k*f)'(x), \end{aligned}$$

which is also called of convolution type in [36]. The relationship between these forms is given by the formula

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }x^\alpha }\,f(x)={\mathbb {D}}_{c,\theta }^\alpha f(x)-f(0)k(x)=:{\mathbb {D}}_{(k)}f(x) \end{aligned}$$

which can be derived as (2.33) in [22] and for more general kernel functions ${\mathbb {D}}_{(k)}f$ is called a generalized fractional derivative in [17]. For further approaches into this direction see [3, 18, 20, 25, 29, 36]. Of particular interest are non-negative locally integrable kernel functions k such that the operator ${\mathbb {D}}_{(k)}$ possesses a right inverse ${\mathbb {I}}_{(k)}$ such that ${\mathbb {D}}_{(k)}{\mathbb {I}}_{(k)}f=f$. Using the theory of complete Bernstein functions and the relationship to the Stieltjes class, it is shown in [17] that this is possible with

$$\begin{aligned} {\mathbb {I}}_{(k)}f(x)=\int _0^x f(y)\,k^*(x-y)\,\hbox {d}y \end{aligned}$$

for locally bounded measurable functions f if $(k,k^*)$ forms a Sonine pair of kernels, i.e. $k*k^*\equiv 1$; cf. also [27, 33, 37]. In this case ${\mathbb {I}}_{(k)}f$ is called a generalized fractional integral of order $\alpha$ and it also holds that ${\mathbb {I}}_{(k)}{\mathbb {D}}_{(k)}f(x)=f(x)-f(0)$ for absolutely continuous functions f.

We will now show that a Sonine kernel $k^*$ may exist for our specific kernel function $k(y)=y^{-\alpha }\theta (\log y)$ working in the more general framework of Bernstein functions and their relation to completely monotone functions. As argued in [9], necessarily the kernel function k should have an integrable singularity at the origin to be a meaningful generalization of a fractional derivative, but we are not going into this discussion. Hence we aim for possible extensions of the list of specific kernel functions given in section 6 of [20]. Our approach works similar to the method laid out in [17] but, to work in full generality, we have to relax the definition of a Sonine pair in the following sense.

Definition 2.8

Given two Borel measures $\eta ,\rho$ on $[0,\infty )$ we say that $(\eta ,\rho )$ forms a generalized Sonine pair if $\eta *\rho =\lambda$, where $\lambda$ denotes Lebesgue measure restricted to $[0,\infty )$.

Theorem 2.9

Let $\widetilde{\psi }$ be a Bernstein function with triplet $[0,0,\phi ]$ and define $\eta$ as the measure on $[0,\infty )$ with $\eta (\{0\})=0$ and density $t\mapsto \phi (t,\infty )$ with respect to Lebesgue measure on $(0,\infty )$, i.e. $d\eta (t)=\phi (t,\infty )\,dt$. Then $\eta$ is a Borel measure and there exists another Borel measure $\rho$ such that $(\eta ,\rho )$ forms a generalized Sonine pair.

Proof

First note that $\eta$ is locally finite, since by the Fubini-Tonelli theorem for all $y>0$ we have

$$\begin{aligned} \eta ([0,y])=\int _{0}^{y}\int _{t}^{\infty }\,\hbox {d}\phi (x)\,\hbox {d}t=\int _{0}^{\infty }\min \{x,y\}\,\hbox {d}\phi (x)<\infty . \end{aligned}$$

Hence $\eta$ is a Borel measure. We have that $\widetilde{\psi }(x)=\int _0^\infty (1-e^{-xt})\,d\phi (t)$ is a Bernstein function and thus $G(x):=\frac{1}{x}\,\widetilde{\psi }(x)$ is completely monotone by Corollary 3.8(iv) in [31]. The tail function of the Lévy measure is a right-continuous, non-increasing function and integration by parts yields the Laplace transform

$$\begin{aligned} G(x)=\int _0^\infty \tfrac{1}{x}(1-e^{-xt})\,\hbox {d}\phi (t)=\int _0^\infty e^{-xt}\phi (t,\infty )\,\hbox {d}t=\widetilde{\eta }(x) \end{aligned}$$

(2.12)

for $x>0$. Further, by Theorem 3.7 in [31], $G^*(x):=1/\widetilde{\psi }(x)$ is completely monotone, since it is the composition of a Bernstein function and the completely monotone function $x\mapsto \frac{1}{x}$. Hence there exists a Borel measure $\rho$ on $[0,\infty )$ such that $G^*(x)=\int _0^\infty e^{-xt}\,d\rho (t)=\widetilde{\rho }(x)$, which serves as the generalized Sonine partner of $\eta$ as follows. We easily calculate the Laplace transform of $\eta *\rho$ as

$$\begin{aligned} \widetilde{\eta }(x)\cdot \widetilde{\rho }(x)=G(x)\cdot G^*(x)=\frac{\widetilde{\psi }(x)}{x}\cdot \frac{1}{\widetilde{\psi }(x)}=\frac{1}{x} \end{aligned}$$

(2.13)

for $x>0$, which is the Laplace transform of the Lebesgue measure $\lambda$ on $[0,\infty )$. Now the assertion directly follows from uniqueness of the Laplace transform for probability measures and the translation principle; see Theorem 1a in chapter XIII.1 of [7]. $\square$

By virtue of Theorem 2.9 we now define a generalized fractional derivative and a corresponding generalized fractional integral of suitable functions $f:[0,\infty )\rightarrow {\mathbb R}$ as

$$\begin{aligned} {\mathbb {D}}_{(\eta )}f(x)=\int _0^x f'(x-y)\,\hbox {d}\eta (y)=(f'*\eta )(x) \end{aligned}$$

(2.14)

for absolutely continuous functions f with locally bounded density $f'$, respectively

$$\begin{aligned} {\mathbb {I}}_{(\rho )}f(x)=\int _0^x f(x-y)\,\hbox {d}\rho (y)=(f*\rho )(x) \end{aligned}$$

(2.15)

for locally bounded measurable functions f.

Lemma 2.10

For absolutely continuous functions $f:[0,\infty )\rightarrow {\mathbb R}$ with locally bounded density $f'$ we have

$$\begin{aligned} \begin{aligned} {\mathbb {D}}_{(\eta )}f(x)&=\frac{\hbox {d}}{\hbox {d}x}\int _0^x f(x-y)\,\hbox {d}\eta (y)-f(0)\,\phi (x,\infty )\\&=(f*\eta )'(x)-f(0)\,\phi (x,\infty ) \end{aligned} \end{aligned}$$

(2.16)

for almost all $x>0$.

We may also use (2.16) as the definition of the generalized fractional derivative which has some advantage, since it can be defined for the broader class of locally bounded measurable functions f due to $\int _0^x f(x-y)\,\hbox {d}\eta (y)=\int _0^x f(x-y)\,\phi (y,\infty )\,dy$ being absolutely continuous.

Proof

Let $k(x)=\phi (x,\infty )$ be the tail function of the Lévy measure. Then by partial integration and the Fubini–Tonelli theorem the Laplace transform on the right-hand side of (2.16) can be written as

$$\begin{aligned}&\int _{0}^{\infty }e^{-sx}(f*k)'(x)\,\hbox {d}x-f(0)\widetilde{k}(s)= s\int _{0}^{\infty }e^{-sx}(f*k)(x)\,\hbox {d}x-f(0)\,\widetilde{k}(s)\\&\quad =s\int _{0}^{\infty }e^{-sx}\int _{0}^{x}f(x-y) k(y)\,\hbox {d}y\,\hbox {d}x-f(0)\,\widetilde{k}(s)\\&\quad =s\int _{0}^{\infty }k(y)\int _{y}^{\infty }e^{-sx}f(x-y)\,\hbox {d}x\,\hbox {d}y-f(0)\,\widetilde{k}(s)\\&\quad =s\,\widetilde{f}(s)\int _{0}^{\infty }e^{-sy}k(y)\,\hbox {d}y-f(0)\,\widetilde{k}(s)\\&\quad =s\,\widetilde{f}(s)\,\widetilde{k}(s)-f(0)\,\widetilde{k}(s) \end{aligned}$$

On the other hand, by the Fubini–Tonelli theorem and partial integration the Laplace transform of (2.14) is given by

$$\begin{aligned}&\int _{0}^{\infty }e^{-sx}\int _{0}^{x}f'(x-y) k(y)\,\hbox {d}y\,\hbox {d}x=\int _{0}^{\infty }k(y)\int _{y}^{\infty }e^{-sx}f'(x-y)\,\hbox {d}x\,\hbox {d}y\\&\quad =\int _{0}^{\infty }k(y)\left( -e^{-sy}f(0)+s\int _{y}^{\infty }e^{-sx}f(x-y)\,\hbox {d}x\right) \,\hbox {d}y\\&\quad =-f(0)\,\widetilde{k}(s)+s\,\widetilde{f}(s)\,\widetilde{k}(s), \end{aligned}$$

which proves the assertion. $\square$

Theorem 2.11

Let the assumptions of Theorem 2.9 hold.

• (a) For absolutely continuous functions f with locally bounded density $f'$ we have ${\mathbb {I}}_{(\rho )}{\mathbb {D}}_{(\eta )}f(x)=f(x)-f(0)$ for all $x>0$.

• (b) For locally bounded measurable functions f we have ${\mathbb {D}}_{(\eta )}{\mathbb {I}}_{(\rho )}f(x)=f(x)$ for almost all $x>0$. If f is continuous then the relation holds for all $x>0$.

Proof

(a) For absolutely continuous functions f with density $f'$ we get

$$\begin{aligned} {\mathbb {I}}_{(\rho )}{\mathbb {D}}_{(\eta )}f(x)&=\int _0^x{\mathbb {D}}_{(\eta )}f(x-y))\,\hbox {d}\rho (y)=\int _0^x (f'*\eta )(x-y)\,\hbox {d}\rho (y)\\&=((f'*\eta *\rho )(x)=(f'*\lambda )(x)=\int _{0}^{x}f'(x-t)\,\hbox {d}t\\&=\int _0^xf'(t)\,\hbox {d}t=f(x)-f(0). \end{aligned}$$

(b) Since ${\mathbb {I}}_{(\rho )}f(0)=0$ and $\int _{0}^{x}f(t)\,\hbox {d}t$ is absolutely continuous, we get

$$\begin{aligned} {\mathbb {D}}_{(\eta )}{\mathbb {I}}_{(\rho )}f(x)&=\frac{\hbox {d}}{\hbox {d}x}\int _0^x{\mathbb {I}}_{(\rho )}f(x-y)\,\hbox {d}\eta (y)=\frac{\hbox {d}}{\hbox {d}x}({\mathbb {I}}_{(\rho )}f*\eta )(x)\\&=\frac{\hbox {d}}{\hbox {d}x}(f*\rho *\eta )(x)=\frac{\hbox {d}}{\hbox {d}x}(f*\lambda )(x)=\frac{\hbox {d}}{\hbox {d}x}\int _{0}^{x}f(x-t)\,\hbox {d}t\\&=\frac{\hbox {d}}{\hbox {d}x}\int _0^xf(t)\,\hbox {d}t=f(x) \end{aligned}$$

for almost all $x>0$. If f is continuous, this relation holds for all $x>0$, since the last equality holds by the fundamental theorem of calculus. $\square$

In our specific context of self-similar Bernstein functions, the semi-fractional integral given by the Sonine partner $\rho$ in (2.15) is also determined by a self-similar Bernstein function as follows.

Lemma 2.12

Let $\widetilde{\psi }$ be a self-similar Bernstein function with respect to $\alpha \in (0,1)$ and $c>1$. Let $\rho$ be the associated Sonine partner from above, then the primitive $G_I^*(x):=\int _0^xG^*(y)\,\hbox {d}y=\int _0^x\widetilde{\rho }(y)\,\hbox {d}y$ is a self-similar Bernstein function with respect to $1-\alpha \in (0,1)$ and $d:=c^{\frac{1-\alpha }{\alpha }}>1$.

Proof

From Lemma 2.2 we get the scaling relation

$$\begin{aligned} G^*(c^{1/\alpha }x)=\frac{1}{\widetilde{\psi }(c^{1/\alpha }x)}=\frac{1}{c\cdot \widetilde{\psi }(x)}=c^{-1}G^*(x)\quad \text { for all }x>0, \end{aligned}$$

which implies $(c^{1/\alpha }\rho )=c^{-1}\cdot \rho$ for the measure $\rho$ in Theorem 2.9. In particular it follows that $\rho (\{0\})=0$ and by the Fubini–Tonelli theorem we get

$$\begin{aligned} G_I^*(x)=&\int _0^xG^*(y)\,\hbox {d}y=\int _0^x\int _{0+}^\infty e^{-yt}\,\hbox {d}\rho (t)\,\hbox {d}y\\ =&\int _{0+}^\infty \frac{1-e^{-xt}}{t}\,\hbox {d}\rho (t)=\int _{0}^\infty (1-e^{-xt})\,\hbox {d}\mu (t), \end{aligned}$$

where the Borel measure $\mu$ on $(0,\infty )$ is given by $\hbox {d}\mu (t):=\frac{1}{t} \hbox {d}\rho (t)$ and integrates $\min \{1,t\}$ as shown in the proof of Theorem 3.2 in [31]; cf. also Proposition 3.5 in [31]. Thus the primitive $G_I^*(x)$ is a Bernstein function and the scaling relation gives us

$$\begin{aligned} G_I^*(c^{1/\alpha }x)&:= \int _{0}^\infty \left( 1-e^{-xc^{1/\alpha }t}\right) \,\hbox {d}\mu (t)=c^{1/\alpha }\int _{0+}^\infty \frac{1-e^{-xc^{1/\alpha }t}}{c^{1/\alpha }\,t}\,\hbox {d}\rho (t)\\&= c^{1/\alpha }\int _{0+}^\infty \frac{1-e^{-xt}}{t}\,d(c^{1/\alpha }\rho )(t)\\&= c^{\frac{1}{\alpha }-1}\int _{0}^\infty (1-e^{-xt})\,\hbox {d}\mu (t)=c^{\frac{1-\alpha }{\alpha }}G_I^*(x). \end{aligned}$$

Now let $d=c^{\frac{1-\alpha }{\alpha }}>1$ then $d^{\frac{1}{1-\alpha }}=c^{1/\alpha }$ and we have $G_I^*(d^{\frac{1}{1-\alpha }}x)=d\cdot G_I^*(x)$ for all $x>0$. Hence $G_I^*$ is a self-similar Bernstein function with respect to $1-\alpha \in (0,1)$ and $d>1$. $\square$

Remark 2.13

By Lemma 2.2 the Lévy measure $\mu$ corresponding to $G_I^*$ is given by $\mu (t,\infty )=t^{\alpha -1}\sigma (\log t)$ for an admissible function $\sigma$ with respect to the parameters $1-\alpha \in (0,1)$ and $d>1$. If this admissible function $\sigma$ is smooth, we get a Sonine pair $(k,k^*)$ in the original sense as follows. Integration by parts yields

$$\begin{aligned} G_I^*(x)=x\int _{0}^\infty \tfrac{1}{x}(1-e^{-xt})\,\hbox {d}\mu (t)=x\int _{0}^\infty e^{-xt}t^{\alpha -1}\sigma (\log t)\,\hbox {d}t \end{aligned}$$

and hence we get as the derivative

$$\begin{aligned} G^*(x)&=\int _{0}^\infty e^{-xt}t^{\alpha -1}\sigma (\log t)\,\hbox {d}t-x\int _{0}^\infty e^{-xt}t^{\alpha }\sigma (\log t)\,\hbox {d}t\\&=\int _{0}^\infty e^{-xt}\left( t^{\alpha -1}\sigma (\log t)-\tfrac{\hbox {d}}{\hbox {d}t}\left( t^{\alpha }\sigma (\log t)\right) \right) \,\hbox {d}t\\&=\int _{0}^\infty e^{-xt}\,t^{\alpha -1}\left( (1-\alpha )\sigma (\log t)-\sigma '(\log t)\right) \,\hbox {d}t\\&=:\int _{0}^\infty e^{-xt}\,k^*(t)\,\hbox {d}t, \end{aligned}$$

where the kernel $k^*$ is non-negative by Lemma A.1. Now the calculation of the Laplace transform in the proof of Theorem 2.9 shows that $(k,k^*)$ is a Sonine pair. Note that in general we cannot expect $k^*$ to be completely monotone as in the approach of [17] with complete Bernstein functions. Nor can we expect that the admissible function $\sigma$ is smooth in general.

3 Discrete Approximation of the Generator

Recall that by (1.10) the semi-fractional derivative operator of order $\alpha \in (0,1)$ is given by the negative generator of the continuous convolution semigroup $(\nu ^{*t})_{t\ge 0}$, where $\nu$ is the semistable distribution with log-characteristic function (1.9). If the tail of the Lévy measure is given by $\phi (t,\infty )=\frac{1}{\Gamma (1-\alpha )}\,t^{-\alpha }$, i.e. the admissible function $\theta \equiv \frac{1}{\Gamma (1-\alpha )}$ is constant, it is well known that the semi-fractional derivative coincides with the ordinary Riemann–Liouville fractional derivative of order $\alpha \in (0,1)$ and can be approximated by means of the Grünwald-Letnikov formula

$$\begin{aligned} \frac{\partial ^{\alpha }}{\partial x^{\alpha }}\,f(x)=\lim _{h\downarrow 0}h^{-\alpha }\sum _{j=0}^{\infty }{\alpha \atopwithdelims ()j}(-1)^{j}f(x-jh) \end{aligned}$$

(3.1)

for functions f in the Sobolev space $W^{2,1}({\mathbb R})$; see section 2.1 in [22] for details. The coefficients in the Grünwald-Letnikov approximation formula appear in a discrete distribution on the positive integers called Sibuya distribution, which first appeared in [32]. A discrete random variable $X_{\alpha }$ on ${\mathbb N}$ is Sibuya distributed with parameter $\alpha \in (0,1)$ if

$$\begin{aligned} {\mathbb {P}}(X_{\alpha }=j)=(-1)^{j-1}{\alpha \atopwithdelims ()j}=(-1)^{j-1}\frac{\alpha (\alpha -1)\cdots (\alpha -j+1)}{j!}\quad \text { for }j\in {\mathbb N}. \end{aligned}$$

(3.2)

For further details and extensions of the Sibuya distribution we refer to [5, 6, 19] and the literature mentioned therein. Using (3.2) we may rewrite (3.1) as

$$\begin{aligned} \frac{\partial ^{\alpha }}{\partial x^{\alpha }}\,f(x)&=\lim _{h\downarrow 0}h^{-\alpha }\sum _{j=0}^{\infty }{\alpha \atopwithdelims ()j}(-1)^{j}f(x-jh)\\&=\lim _{h\downarrow 0}h^{-\alpha }\left( f(x)-\sum _{j=1}^{\infty }{\mathbb {P}}(X_{\alpha }=j)f(x-jh)\right) \\&=\lim _{h\downarrow 0}h^{-\alpha }\left( f(x)-\int _{{\mathbb R}}f(x-hy)\,d{\mathbb {P}}_{X_{\alpha }}(y)\right) \\&=\lim _{h\downarrow 0}h^{-\alpha }\left( f*\left( \varepsilon _{0}-{\mathbb {P}}_{hX_{\alpha }}\right) \right) (x) \end{aligned}$$

which shows that the Grünwald–Letnikov formula is in fact a discrete approximation of the generator. For further relations of the Sibuya distribution to fractional diffusion equations see [23, 24].

Our aim is to generalize this formula for semi-fractional derivatives by means of the corresponding self-similar Bernstein functions. Note that for the Bernstein function $\widetilde{\psi }(x)=x^{\alpha }$ the nominator on the right-hand side of (3.2) is given by $\widetilde{\psi }^{(j)}(1)$ and hence we may define a semi-fractional Sibuya distribution in the following way.

Definition 3.1

Given an admissible function $\theta$ with respect to $\alpha \in (0,1)$ and $c>1$, a semi-fractional Sibuya distributed random variable $X_{\theta }$ on ${\mathbb N}$ is given by

$$\begin{aligned} {\mathbb {P}}(X_{\theta }=j)=\frac{(-1)^{j-1}}{j!}\,\frac{\widetilde{\psi }^{(j)}(1)}{\widetilde{\psi }(1)}\quad \text { for }j\in {\mathbb N}, \end{aligned}$$

(3.3)

where $\widetilde{\psi }$ is the corresponding self-similar Bernstein function.

Clearly, the expression in (3.3) is non-negative by (1.1). Moreover, by monotone convergence and a Taylor series approach justified by Proposition 3.6 in [31] we get

$$\begin{aligned} \sum _{j=1}^{\infty }\frac{(-1)^{j-1}}{j!}\,\frac{\widetilde{\psi }^{(j)}(1)}{\widetilde{\psi }(1)}&=\frac{1}{\widetilde{\psi }(1)}\sum _{j=1}^{\infty }\frac{\widetilde{\psi }^{(j)}(1)}{j!}\lim _{\varepsilon \downarrow 0}(\varepsilon -1)^{j-1}\\&=\frac{1}{\widetilde{\psi }(1)}\lim _{\varepsilon \downarrow 0}\frac{1}{\varepsilon -1}\left( \sum _{j=0}^{\infty }\frac{\widetilde{\psi }^{(j)}(1)}{j!}(\varepsilon -1)^{j}-\widetilde{\psi }(1)\right) \\&=\frac{1}{\widetilde{\psi }(1)}\lim _{\varepsilon \downarrow 0}\frac{1}{\varepsilon -1}\left( \widetilde{\psi }(\varepsilon )-\widetilde{\psi }(1)\right) =\frac{\widetilde{\psi }(1)-\widetilde{\psi }(0)}{\widetilde{\psi }(1)}=1. \end{aligned}$$

This shows that indeed (3.3) defines a proper distribution on ${\mathbb N}$ with pgf

$$\begin{aligned} G(z)&=\sum _{j=1}^{\infty }{\mathbb {P}}(X_{\theta }=j)\, z^{j} = -\,\frac{1}{\widetilde{\psi }(1)}\sum _{j=1}^{\infty }\frac{\widetilde{\psi }^{(j)}(1)}{j!}\,(-z)^{j}\\&=1-\frac{1}{\widetilde{\psi }(1)}\sum _{j=0}^{\infty }\frac{\widetilde{\psi }^{(j)}(1)}{j!}\,((1-z)-1)^{j}=1-\frac{\widetilde{\psi }(1-z)}{\widetilde{\psi }(1)} \end{aligned}$$

for $|z|\le 1$. Note that in the above arguments self-similarity of the Bernstein function is not needed. Hence (3.3) defines a proper distribution on ${\mathbb N}$ for every Bernstein function $\widetilde{\psi }$ with $\widetilde{\psi }(0)=0$.

Now let $\theta$ be a smooth admissible function having Fourier series representation

$$\begin{aligned} \theta (x)=\sum _{k=-\infty }^{\infty }c_{k} e^{ik\tilde{c} x}\quad \text { with }\quad \tilde{c}=\tfrac{2\pi \alpha }{\log c}. \end{aligned}$$

Then by Theorem 3.1 in [13] the corresponding self-similar Bernstein function can be expressed in terms of the log-characteristic function $\psi$ as

$$\begin{aligned} \widetilde{\psi }(x)=-\psi (ix)=\sum _{k=-\infty }^{\infty }\omega _{k} x^{\alpha -ik\tilde{c}}\quad \text { with }\quad \omega _{k}=c_{k}\Gamma (ik\tilde{c}-\alpha +1) \end{aligned}$$

and hence for $j\in {\mathbb N}_{0}$ we have

$$\begin{aligned} \frac{\widetilde{\psi }^{(j)}(1)}{j!}=\sum _{k=-\infty }^{\infty }\omega _{k}{\alpha -ik\tilde{c}\atopwithdelims ()j}. \end{aligned}$$

Note that these coefficients appear in a Grünwald–Letnikov-type approximation of the semi-fractional derivative given in Theorem 4.1 of [13] which enables us to prove the following approximation formula.

Theorem 3.2

Let $\theta$ be a smooth admissible function with respect to $\alpha \in (0,1)$ and $c>1$ with corresponding self-similar Bernstein function $\widetilde{\psi }$ and semi-fractional Sibuya distributed random variable $X_{\theta }$ given by (3.3). Then for $f\in W^{2,1}({\mathbb R})$ the semi-fractional derivative can be approximated along the subsequence $h_{m}=c^{-m/\alpha }$ by

$$\begin{aligned} \frac{\partial ^{\alpha }}{\partial _{c,\theta } x^{\alpha }}\,f(x)=\lim _{m\rightarrow \infty }\widetilde{\psi }(h_{m}^{-1})\left( f*\left( \varepsilon _{0}-{\mathbb {P}}_{h_{m }X_{\theta }}\right) \right) (x). \end{aligned}$$

Proof

First note that $h_{m}^{ik\tilde{c}}=c^{-imk\tilde{c}/\alpha }=e^{-2\pi imk}=1$ for $\tilde{c}=\tfrac{2\pi \alpha }{\log c}$ and by self-similarity we have $\widetilde{\psi }(h_{m}^{-1})=h_{m}^{-\alpha }\widetilde{\psi }(1)$ for all $m\in {\mathbb N}$. Hence we get

$$\begin{aligned}&\widetilde{\psi }(h_{m}^{-1})\left( f*\left( \varepsilon _{0}-{\mathbb {P}}_{h_{m }X_{\theta }}\right) \right) (x) =h_{m}^{-\alpha }\widetilde{\psi }(1)\left( f(x)-\int _{{\mathbb R}}f(x-h_{m}y)\,d{\mathbb {P}}_{X_{\theta }}(y)\right) \\&\quad =h_{m}^{-\alpha }\widetilde{\psi }(1)\left( f(x)-\sum _{j=1}^{\infty }\frac{(-1)^{j-1}}{j!}\,\frac{\widetilde{\psi }^{(j)}(1)}{\widetilde{\psi }(1)}\,f(x-jh_{m})\right) \\&\quad =h_{m}^{-\alpha }\left( \sum _{k=-\infty }^{\infty }\omega _{k}f(x)+\sum _{j=1}^{\infty }(-1)^{j}\sum _{k=-\infty }^{\infty }\omega _{k}{\alpha -ik\tilde{c}\atopwithdelims ()j}f(x-jh_{m})\right) \\&\quad =h_{m}^{-\alpha }\sum _{j=0}^{\infty }(-1)^{j}\sum _{k=-\infty }^{\infty }\omega _{k}h_{m}^{ik\tilde{c}} {\alpha -ik\tilde{c}\atopwithdelims ()j}f(x-jh_{m})\rightarrow \frac{\partial ^{\alpha }}{\partial _{c,\theta } x^{\alpha }}\,f(x), \end{aligned}$$

where the last convergence follows from Theorem 4.1 in [13]. $\square$

4 Space-Time Duality for Semi-Fractional Diffusions

We first give a sufficient condition for an inverse function to be a Bernstein function.

Lemma 4.1

Let $f:(0,\infty )\rightarrow (0,\infty )$ be a $C^\infty (0,\infty )$-function such that $f'$ is a Bernstein function and $f^{(n)}(x)\not =0$ for all $x>0$ and $n\in {\mathbb N}$. Then its inverse $f^{-1}$ is a Bernstein function with $(f^{-1})^{(n)}(x)\not =0$ for all $x>0$ and $n\in {\mathbb N}$.

Proof

We know that $f^{-1}(x)>0$ and $(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}>0$ for all $x>0$. Moreover, as in Remark A.3 of the Appendix $(f^{-1})''(x)=-\,\frac{f''(f^{-1}(x))}{(f'(f^{-1}(x)))^3}<0$ for all $x>0$. For $n\ge 3$ we inductively use the formula for $(f^{-1})^{(n)}$ given in Lemma A.2 of the Appendix. By induction we have

$$\begin{aligned} {{\,\mathrm{sign}\,}}\left( \prod _{j=1}^{n-1}\left( \frac{(f^{-1})^{(j)}(x)}{j!}\right) ^{k_j}\right) =\prod _{j=1}^{n-1}(-1)^{(j-1)k_j}=(-1)^{\sum _{j=1}^{n-1}(j-1)k_j} \end{aligned}$$

and ${{\,\mathrm{sign}\,}}(f^{(k_1+\cdots +k_{n-1})}(f^{-1}(x)))=(-1)^{k_1+\cdots +k_{n-1}}$ since $k_1+\cdots +k_{n-1}\ge 2$ for $n\ge 3$ as $k_1+2k_2\cdots +(n-1)k_{n-1}=n$. This shows that

$$\begin{aligned} {{\,\mathrm{sign}\,}}\left( f^{(k_1+\cdots +k_{n-1})}(f^{-1}(x))\prod _{j=1}^{n-1}\left( \frac{(f^{-1})^{(j)}(x)}{j!}\right) ^{k_j}\right) =(-1)^{\sum _{j=1}^{n-1}jk_j}=(-1)^n \end{aligned}$$

for every summand of the formula in Lemma A.2. Thus ${{\,\mathrm{sign}\,}}((f^{-1})^{(n)}(x))=(-1)^{n-1}$ for all $x>0$ and $n\ge 3$ showing that $f^{-1}$ is a Bernstein function. $\square$

We want to apply Lemma 4.1 to the function

$$\begin{aligned} \zeta (x)=\psi (-ix)=\int _{-\infty }^0\left( e^{xy}-1-xy\right) \,\hbox {d}\phi (y)>0\quad \text { for }x>0, \end{aligned}$$

(4.1)

where $\phi$ is the semistable Lévy measure for $\alpha \in (1,2)$ from (1.8) concentrated on the negative axis and $\psi$ is the corresponding log-characteristic function from (1.9). Note that $\zeta (x)\rightarrow \infty$ as $x\rightarrow \infty$ due to the tail behavior in (1.8) and by dominated convergence we have

$$\begin{aligned} \zeta '(x)=\int _{-\infty }^0y\left( e^{xy}-1\right) \,\hbox {d}\phi (y)=\int _0^\infty \left( 1-e^{-xy}\right) y\,\hbox {d}\phi (-y). \end{aligned}$$

Since $\phi$ integrates $\min \{1,y^2\}$ and fulfills (1.8), the measure $\mu$ with $\hbox {d}\mu (y)=y\,\hbox {d}\phi (-y)$ integrates $\min \{1,y\}$ and thus $\zeta '$ is a Bernstein function. This also follows from the higher order derivatives

$$\begin{aligned} \zeta ^{(n)}(x)=\int _{-\infty }^0y^ne^{xy}\,\hbox {d}\phi (y){\left\{ \begin{array}{ll} >0 &{} \text { if}\; n\ge 2 \;\text {is even,}\\ <0 &{} \text { if}\; n\ge 3\;\text { is odd,}\end{array}\right. } \end{aligned}$$

which additionally shows that these derivatives do not vanish. Thus from Lemma 4.1 we conclude that the inverse $\zeta ^{-1}$ is a Bernstein function. This will be the key to solve an open problem concerning the following result on space-time duality from [12].

According to [30, Example 28.2], the semistable Lévy process $(X_t)_{t\ge 0}$ with $P\{X_t\in A\}=\nu ^{*t}(A)$ for Borel sets $A\in \mathcal B({\mathbb R})$, where $\nu$ is the semistable distribution with log-characteristic function $\psi$ from (1.9), possesses $C^\infty ({\mathbb R})$-densities $x\mapsto p(x,t)$ for every $t>0$ with $P\{X_t\in A\}=\int _Ap(x,t)\,\hbox {d}x$ and for $t=0$ we may write $p(x,0)=\delta (x)$ corresponding to $X_{0}=0$ almost surely. It is shown in [13] that these densities are the point source solution to the semi-fractional diffusion equation

$$\begin{aligned} \frac{\partial ^\alpha }{\partial _{c,\theta }(-x)^\alpha }\,p(x,t)=\frac{\partial }{\partial t}\,p(x,t) \end{aligned}$$

(4.2)

with the negative semi-fractional derivative of order $\alpha$ from (1.14) acting on the space variable. Since the semi-fractional derivative is a non-local operator, this equation is hard to interpret from a physical point of view, whereas non-locality in time may correspond to long memory effects [10]. As a generalization of a space–time duality result for fractional diffusions [4, 11] based on a corresponding result for stable densities by Zolotarev [39, 40], it was shown in [12] that space-time duality may also hold for semi-fractional diffusions. Theorem 3.3 in [12] states that for $x>0$ and $t>0$ we have $p(x,t)=\alpha ^{-1}h(x,t)$, where h(x, t) is the point source solution to the semi-fractional differential equation

$$\begin{aligned} \frac{\partial ^{1/\alpha }}{\partial _{\hbox {d},\tau }t^{1/\alpha }}\,h(x,t)+\frac{\partial }{\partial x}\,h(x,t)=t^{-1/\alpha }\varrho (\log t)\,\delta (x) \end{aligned}$$

(4.3)

with a semi-fractional derivative of order $1/\alpha$ acting on the time variable, provided that $\tau$ and $\varrho$ are admissible functions with respect to $1/\alpha \in (\frac{1}{2},1)$ and $d=c^{1/\alpha }>1$ which remained an open problem in [12]. We will now show that indeed $\tau$ is admissible and $\varrho (x)=-\alpha \tau '(x)$, provided that $\tau$ is smooth. Note that in general $\varrho$ will not be admissible as conjectured in [12], but we will justify the inhomogeneity in (4.3) by different arguments. In [12] the function $\tau$ appears in the following way. The above inverse $\zeta ^{-1}$ is called $\xi$ in [12] and its existence is shown in Lemma 4.1 of [12]. It was further shown in Lemma 4.2 of [12] that $\xi (t)=t^{1/\alpha }g(\log t)$ for a continuously differentiable and $\log (c)$-periodic function g. Since we now know that $\xi =\zeta ^{-1}$ is a Bernstein function, in fact g is a $C^\infty ({\mathbb R})$-function. Since g is a smooth $\log (c)$-periodic function, it is representable by its Fourier series

$$\begin{aligned} g(x)=\sum _{n\in {\mathbb Z}}d_ne^{-in\tilde{d}x}\quad \text { with }\tilde{d}=\frac{2\pi }{\log c}=\frac{2\pi \frac{1}{\alpha }}{\log d}\text { for }d=c^{1/\alpha }, \end{aligned}$$

(4.4)

where by Lemma 1 in §12 of [2] we have $|d_{n}|\le C\cdot e^{-\frac{\pi }{2}\,|n|\tilde{d}}$ for some $C>0$ and all $n\in {\mathbb Z}$. If we require a little more quality, namely that the Fourier coefficients even decay as

$$\begin{aligned} |d_{n}|\le C\cdot e^{-\frac{\pi }{2}\,|n|\tilde{d}}|n|^{-\frac{3}{2}-\frac{1}{\alpha }-\varepsilon }\quad \text { for some} \varepsilon >0\; \text {and all}\; n\in {\mathbb Z}{\setminus }\{0\}, \end{aligned}$$

(4.5)

then we can define $\tau$ by the Fourier series

$$\begin{aligned} \tau (x)=\sum _{n\in {\mathbb Z}}\frac{d_n}{\Gamma (in\tilde{d}-\frac{1}{\alpha }+1)}\,e^{-in\tilde{d}x}. \end{aligned}$$

(4.6)

Lemma 4.2

If (4.5) holds, then the function $\tau$ in (4.6) is well-defined and a smooth admissible function with respect to $1/\alpha \in (\frac{1}{2},1)$ and $d=c^{1/\alpha }$. Moreover, $\zeta ^{-1}$ is a self-similar Bernstein function with respect to the same parameters.

Proof

As shown above $\xi =\zeta ^{-1}$ is a Bernstein function and with $d=c^{1/\alpha }$ and $\xi (t)=t^{1/\alpha }g(\log t)$ for a $\log (c)$-periodic function g we get

$$\begin{aligned} d\cdot \xi (t)=(ct)^{1/\alpha }g(\log (ct))=\xi (ct)=\xi (d^\alpha t) \end{aligned}$$

showing that $\xi$ is a self-similar Bernstein function with respect to $1/\alpha \in (\frac{1}{2},1)$ and $d=c^{1/\alpha }$. By Lemma 2.2 we have

$$\begin{aligned} \xi (x)=\int _0^\infty \left( 1-e^{-xy}\right) \,\hbox {d}\mu (y), \end{aligned}$$

(4.7)

where $\mu$ is a semistable Lévy measure with $\mu (t,\infty )=t^{-1/\alpha }\tau (\log t)$ for an admissible function $\tau$ with respect to $1/\alpha \in (\frac{1}{2},1)$ and $d=c^{1/\alpha }$. It remains to show that $\tau$ is indeed the function we are looking for. Let us assume for a moment that $\tau$ is smooth and thus admits the Fourier series

$$\begin{aligned} \tau (x)=\sum _{n\in {\mathbb Z}}a_n\,e^{-in\tilde{d}x}\quad \text { with }\tilde{d}=\frac{2\pi }{\log c}=\frac{2\pi \frac{1}{\alpha }}{\log d}. \end{aligned}$$

In this case the function $\gamma$ appearing in Lemma 2.2 fulfills

$$\begin{aligned} \gamma (-x)=e^{-\frac{1}{\alpha }\,x}\xi (e^x)=g(x) \end{aligned}$$

and by (2.6) has the Fourier series representation

$$\begin{aligned} g(x)=\gamma (-x)=\sum _{n\in {\mathbb Z}}a_n\Gamma (in\tilde{d}-\tfrac{1}{\alpha }+1)\,e^{-in\tilde{d}x}. \end{aligned}$$

A comparison with (4.4) and uniqueness of the Fourier coefficients shows that $a_n$ coincides with $d_n/\Gamma (in\tilde{d}-\tfrac{1}{\alpha }+1)$ and thus $\tau$ is indeed the function in (4.6). Finally, we have to show that the series in (4.6) converges. Using the asymptotic behavior of the gamma function in Corollary 1.4.4 of [1], the Fourier coefficients fulfill

$$\begin{aligned} \left| \frac{d_n}{\Gamma (in\tilde{d}-\frac{1}{\alpha }+1)}\right| \le K |d_{n}|\cdot |n|^{-\frac{1}{2}+\frac{1}{\alpha }} e^{\frac{\pi }{2}\,|n|\tilde{d}} \end{aligned}$$

for a constant $K>0$ and all $n\in {\mathbb Z}{\setminus }\{0\}$. According to our assumption (4.5) we obtain

$$\begin{aligned} \left| \frac{d_n}{\Gamma (in\tilde{d}-\frac{1}{\alpha }+1)}\right| \le KC|n|^{-2-\varepsilon } \end{aligned}$$

for some $\varepsilon >0$ and all $n\in {\mathbb Z}{\setminus }\{0\}$ showing that the series in (4.6) converges and the resulting function is continuously differentiable by Theorem 2.6 in [8]. $\square$

Admissibility of $\tau$ in combination with Theorem 3.3 in [12] finally enables us to completely solve space-time duality for semi-fractional diffusions. The Eq. (4.3) is derived in [12] by Laplace inversion of the equation

$$\begin{aligned} \xi (s)\widetilde{h}(x,s)-\frac{1}{s}\,\xi (s)\delta (x)+\frac{\partial }{\partial x}\,\widetilde{h}(x,s)=-\frac{1}{s}\,\frac{f(s)}{s+f(s)}\,\xi (s)\delta (x), \end{aligned}$$

(4.8)

where f is defined in the proof of Theorem 3.1 in [12] as

$$\begin{aligned} f(s)=\frac{1}{\alpha }\,\xi (s)^\alpha m'(\log \xi (s)) \end{aligned}$$

(4.9)

and m is a $\log (c^{1/\alpha })$-periodic function given by

$$\begin{aligned} \zeta (s)=x^\alpha m(\log x). \end{aligned}$$

(4.10)

Theorem 4.3

Assume that (4.5) holds, $\tau$ is given as in Lemma 4.2 and define $\varrho (t)=-\alpha \tau '(t)$. Then the point source solutions p(x, t) of the semi-fractional diffusion Eq. (4.2) of order $\alpha \in (1,2)$ in space and h(x, t) of the semi-fractional Eq. (4.3) of order $1/\alpha \in (\frac{1}{2},1)$ in time are equivalent, i.e. $p(x,t)=\alpha ^{-1}h(x,t)$ for all $x>0$ and $t>0$.

Proof

As shown in [12] it remains to carry out Laplace inversion of (4.8). Clearly, $\frac{\partial }{\partial x}\,\widetilde{h}(x,s)$ is the Laplace transform of $\frac{\partial }{\partial x}\,h(x,t)$. Moreover, the first part on the left-hand side of (4.8) is the Laplace transform of the semi-fractional derivative in time $\frac{\partial ^{1/\alpha }}{\partial _{d,\tau }t^{1/\alpha }}\,h(x,t)$ as argued in [12]. Since both derivatives are continuous functions of the time variable, Laplace inversion of the left-hand side of (4.8) yields the sum of these derivatives.

We may rewrite the right-hand side of (4.8) as follows. From (4.10) we obtain

$$\begin{aligned} \zeta '(t)=\alpha \,x^{\alpha -1}m(\log x)+x^{\alpha -1}m'(\log x) \end{aligned}$$

and thus we have by (4.10) and $\zeta (\xi (s))=s$

$$\begin{aligned} \zeta '(\xi (s))=\alpha \,\frac{1}{\xi (s)}\,s+\frac{1}{\xi (s)}\,\xi (s)^{\alpha }m'(\log \xi (s))=\alpha \,\frac{1}{\xi (s)}\,s+\alpha \,\frac{1}{\xi (s)}\,f(s), \end{aligned}$$

where the last equality follows from (4.9). This shows that

$$\begin{aligned} f(s)=\frac{1}{\alpha }\,\zeta '(\xi (s))\,\xi (s)-s \end{aligned}$$

and the right-hand side of (4.8) can be rewritten as

$$\begin{aligned} -\frac{1}{s}\,\frac{f(s)}{s+f(s)}\,\xi (s)\delta (x)=-\frac{\alpha }{s}\,\frac{f(s)}{\zeta '(\xi (s))}\,\delta (x)=-\frac{1}{s}\,\xi (s)\delta (x)+\alpha \,\frac{1}{\zeta '(\xi (s))}\,\delta (x). \end{aligned}$$

(4.11)

Note that the first part $-\frac{1}{s}\,\xi (s)\delta (x)$ also appears on the left-hand side of (4.8); cf. Remark 4.4. Moreover, since $\frac{1}{s}$ is the Laplace transform of $1_{(0,\infty )}(t)$, the function $\frac{1}{s}\,\xi (s)$ is the Laplace transform of the Riemann–Liouville semi-fractional derivative of $1_{(0,\infty )}(t)$ which is

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\int _0^t 1_{(0,\infty )}(t-s)\,s^{-1/\alpha }\tau (\log (s))\,\hbox {d}s=t^{-1/\alpha }\tau (\log (t)). \end{aligned}$$

Now from $\zeta (\xi (s))=s$ we get $\zeta '(\xi (s))\cdot \xi '(s)=1$ and hence it follows from (4.7) and integration by parts

$$\begin{aligned} \frac{1}{\zeta '(\xi (s))}&= \xi '(s)=\int _0^\infty e^{-st}t\,\hbox {d}\mu (t)\\&= \left[ -e^{-st}t^{1-1/\alpha }\tau (\log t)\right] _{t=0}^\infty +\int _0^\infty \left( -s\,e^{-st}t+e^{-st}\right) t^{-1/\alpha }\tau (\log t)\,\hbox {d}t\\&= -s\int _0^\infty e^{-st}t^{1-1/\alpha }\tau (\log t)\,\hbox {d}t+\int _0^\infty e^{-st}t^{-1/\alpha }\tau (\log t)\,\hbox {d}t \end{aligned}$$

which is the Laplace transform of

$$\begin{aligned}&-\frac{\hbox {d}}{\hbox {d}t}\left( t^{1-1/\alpha }\tau (\log t)\right) +t^{-1/\alpha }\tau (\log t)\\&\quad = -(1-\tfrac{1}{\alpha })t^{-1/\alpha }\tau (\log t)-t^{-1/\alpha }\tau '(\log t)+t^{-1/\alpha }\tau (\log t)\\&\quad =t^{-1/\alpha }\left( \tfrac{1}{\alpha }\,\tau (\log t)-\tau '(\log t)\right) . \end{aligned}$$

Putting things together, due to continuous differentiability of $\tau$ proven in Lemma 4.2, Laplace inversion of the right-hand side of (4.8) yields

$$\begin{aligned} \left( -t^{-1/\alpha }\tau (\log t)+\alpha t^{-1/\alpha }\left( \tfrac{1}{\alpha }\,\tau (\log t)-\tau '(\log t)\right) \right) \delta (x)=-\alpha t^{-1/\alpha }\tau '(\log t)\delta (x) \end{aligned}$$

concluding the proof. $\square$

Remark 4.4

Note that in the stable case we have that $\tau$ is constant and thus $\tau '\equiv 0$. Hence we recover space-time duality for fractional diffusions in [11] as a special case. Further note that with (4.11) the term $-\frac{1}{s}\,\xi (s)\delta (x)$ appears on both sides of (4.8) and can be cancelled. After cancellation the first part on the left-hand side of (4.8) is the Laplace transform of the Riemann–Liouville semi-fractional derivative of h(x, t) and as in the proof of Theorem 4.3 we may rewrite (4.3) as

$$\begin{aligned} {\mathbb {D}}_{d,\tau }^{1/\alpha }h(x,t)+\frac{\partial }{\partial x}\,h(x,t)=t^{-1/\alpha }\left( \tau (\log t)-\alpha \tau '(\log t) \right) \delta (x), \end{aligned}$$

where the semi-fractional derivative acts on the time variable and now the inhomogeniety on the right-hand side is non-negative by Lemma A.1.

Appendix

We first show an elementary result connecting admissible functions with the representation in Corollary 7.4.4 of [21].

Lemma A.1

Let $\theta :{\mathbb R}\rightarrow (0,\infty )$ be a periodic function and $\alpha >0$. Then the following statements are equivalent.

• (i) $\theta (y+\delta )\le e^{\alpha \delta }\theta (y)$ for all $y>0$ and $\delta >0$.

• (ii) $\theta (y+\delta )\le e^{\alpha \delta }\theta (y)$ for all $y\in {\mathbb R}$ and $\delta \ge 0$.

• (iii) $\theta (y-\delta )\ge e^{-\alpha \delta }\theta (y)$ for all $y\in {\mathbb R}$ and $\delta \ge 0$.

• (iv) The mapping $t\mapsto t^{-\alpha }\theta (\log t)$ is non-increasing for $t>0$.

If $\theta$ is additionally differentiable, then each statement (i)–(iv) is equivalent to

• (v) $\theta '(y)\le \alpha \,\theta (y)$ for all $y\in {\mathbb R}$.

Proof

“(i)$\Rightarrow$(ii)”: Note that (ii) is trivially true for $\delta =0$. If $\theta$ has period $p>0$ and $y\le 0$, choose $k\in {\mathbb N}$ such that $y+kp>0$. Then $\theta (y+\delta )=\theta (y+kp+\delta )\le e^{\alpha \delta }\theta (y+kp)=e^{\alpha \delta }\theta (y)$.

“(ii)$\Rightarrow$(iii)”: Write $z=y-\delta$ then $\theta (y-\delta )=\theta (z)\ge e^{-\alpha \delta }\theta (z+\delta )=e^{-\alpha \delta }\theta (y)$.

“(iii)$\Rightarrow$(iv)”: For $0<s<t$ write $t=e^y$ and $s=e^{y-\delta }$ for some $y\in {\mathbb R}$ and $\delta >0$. Then $s^{-\alpha }\theta (\log s)=e^{-\alpha (y-\delta )}\theta (y-\delta )\ge e^{-\alpha y}\theta (y)=t^{-\alpha }\theta (\log t)$.

“(iv)$\Rightarrow$(i)”: Write $y=\log t$ and $\delta =\log \gamma$ for some $t,\gamma >1$. Then $\theta (y+\delta )=\theta (\log (\gamma t))=(\gamma t)^{\alpha }(\gamma t)^{-\alpha }\theta (\log (\gamma t))\le (\gamma t)^{\alpha }t^{-\alpha }\theta (\log t)=e^{\alpha \delta }\theta (y)$.

“(ii)$\Rightarrow$(v)”: For fixed $y\in {\mathbb R}$ we have $\frac{\theta (y+\delta )-\theta (y)}{\delta }\le \frac{e^{\alpha \delta }-1}{\delta }\,\theta (y)$ for all $\delta >0$ and for differentiable $\theta$ as $\delta \downarrow 0$ we get $\theta '(y)\le \alpha \,\theta (y)$.

“(v)$\Rightarrow$(iv)”: We have $\frac{d}{dt}(t^{-\alpha }\theta (\log t))=t^{-\alpha -1}(-\alpha \theta (\log t)+\theta '(\log t))\le 0$. $\square$

Now we derive a formula for higher order derivatives of inverse functions used in Sect. 3 for which we couldn’t find a suitable reference.

Lemma A.2

Let $A,B\subseteq {\mathbb R}$ be open and let $f:A\rightarrow B$ be an invertible $C^n(A)$-function for some $n\in {\mathbb N}$ such that $f'(x)\not =0$ for all $x\in A$. Then $f^{-1}:B\rightarrow A$ is of class $C^n(B)$ and for $n\ge 2$ we have

$$\begin{aligned} (f^{-1})^{(n)}(x)=-\,\frac{1}{f'(f^{-1}(x))}&\sum _{k_1+2k_2+\cdots +(n-1)k_{n-1}=n}\frac{n!}{k_1!\cdots k_{n-1}!}\,f^{(k_1+\cdots +k_{n-1})}(f^{-1}(x))\\&\qquad \qquad \cdot \prod _{j=1}^{n-1}\left( \frac{(f^{-1})^{(j)}(x)}{j!}\right) ^{k_j}. \end{aligned}$$

Remark A.3

For $n=1$ it is well known that $(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}$. For $n=2$ we have $k_1=2$ in Lemma A.3 so that

$$\begin{aligned} (f^{-1})''(x)=-\,\frac{1}{f'(f^{-1}(x))}\,f''(f^{-1}(x))\cdot ((f^{-1})'(x))^2=-\,\frac{f''(f^{-1}(x))}{(f'(f^{-1}(x)))^3}. \end{aligned}$$

Iterating this procedure leads to a formula for $(f^{-1})^{(n)}$ given in [38] not involving the derivatives of lower order $(f^{-1})',\ldots ,(f^{-1})^{(n-1)}$.

Proof of Lemma A.2

We use Faà di Bruno’s formula of higher order chain rule

$$\begin{aligned} \frac{\hbox {d}^n}{\hbox {d}x^n}\,f(g(x))=&\sum _{k_1+2k_2+\cdots +nk_n=n}\frac{n!}{k_1!\cdots k_{n}!}\,f^{(k_1+\cdots +k_{n})}(g(x))\prod _{j=1}^{n}\left( \frac{(g^{(j)}(x)}{j!}\right) ^{k_j}. \end{aligned}$$

For $j=n$ we must have $k_n=1$ and $k_1=\cdots =k_{n-1}=0$, otherwise $k_n=0$ if $k_j\ge 1$ for some $j\in \{1,\ldots ,n-1\}$. Hence we get

$$\begin{aligned} \frac{\hbox {d}^n}{\hbox {d}x^n}\,f(g(x))=&\sum _{k_1+2k_2+\cdots +(n-1)k_{n-1}=n}\frac{n!}{k_1!\cdots k_{n}!}\,f^{(k_1+\cdots +k_{n-1})}(g(x))\prod _{j=1}^{n-1}\left( \frac{(g^{(j)}(x)}{j!}\right) ^{k_j}\\&\qquad \qquad +f'(g(x))\,g^{(n)}(x). \end{aligned}$$

If $g=f^{-1}$, we know that $\frac{\hbox {d}^n}{\hbox {d}x^n}\,f(f^{-1}(x))=0$ for $n\ge 2$ which directly leads to the stated formula for $(f^{-1})^{(n)}$. $\square$