A class of nowhere differentiable functions satisfying some concavity type estimate

TL;DR

This paper introduces a class of continuous, nowhere differentiable functions characterized by a concavity-type estimate on second-order differences, exploring their geometric, analytic, and functional series properties.

Contribution

It defines and characterizes the class P of functions via geometric inequalities and difference estimates, linking it to nowhere differentiability and functional series.

Findings

Functions in P are nowhere differentiable.

A geometric inequality characterizes functions in P.

Explicit examples of functions in P are constructed from series.

Abstract

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this function. As its application, we study the behavior of the Hamilton Jacobi flow starting from a function in P. A connection between P and some functional series is also investigated. In terms of second-order central differences, we give a necessary and sufficient condition so that a function given by the series belongs to P. This enables us to construct a large number of examples of functions in P through an explicit formula.

Equations

$$\delta^{+}_{n,k}(y;f)=\frac{f\bigl{(}\frac{k+1}{r^{n}}\bigr{)}-f\bigl{(}\frac{k+y}{r^{n}}\bigr{)}}{\frac{1-y}{r^{n}}},\quad\delta^{-}_{n,k}(y;f)=\frac{f\bigl{(}\frac{k+y}{r^{n}}\bigr{)}-f\bigl{(}\frac{k}{r^{n}}\bigr{)}}{\frac{y}{r^{n}}}.$$

$${\delta }_{n,k}^{+}(y;f)-{\delta }_{n,k}^{-}(y;f)\le -c$$

$${\Delta }_{n,k}(y;f)\le -2c{r}^n,$$

$${\Delta }_{n,k}(y;f)=2r^n({\delta }_{n,k}^{+}(y;f)-{\delta }_{n,k}^{-}(y;f)).$$

$$q_f(t,x;z)=f(z)+\frac{1}{2t}(x-z{)}^2,(t,x,z)\in (0,\infty )\times \mathbb{R}\times \mathbb{R}.$$

$$q_f\left(t,x;\frac{k+y}{r^n}\right)\ge \min \left\{q_f\left(t,x;\frac{k}{r^n}\right),q_f\left(t,x;\frac{k+1}{r^n}\right)\right\},x\in \mathbb{R}.$$

$$U_{\psi }(x)=\sum _{j=0}^{\infty }\frac{1}{r^j}\psi (r^{j}x),x\in \mathbb{R}.$$

$${\Delta }_{n,k}(y;U_{\psi })=\sum _{j=0}^{n-1}{r}^j{\Delta }_{n-j,k}(y;\psi )-\frac{2r^n}{y(1-y)}{U}_{\psi }(y),$$

$${\tau }_r(x)=U_d(x)=\sum _{j=0}^{\infty }\frac{1}{r^j}d(r^{j}x),x\in \mathbb{R},$$

$$d(x)=\min \{|x-z||z\in \mathbb{Z}\},x\in \mathbb{R}.$$

$$H_{t}f(x)=\underset{z\in \mathbb{R}}{\inf }{q}_f(t,x;z),(t,x)\in (0,\infty )\times \mathbb{R}.$$

$$\mathbb{A}:=\{(n,k,y)\mid n\in {\mathbb{N}}_0,k\in \{0,1,2,3,\dots ,r^n-1\},y\in (0,1)\}.$$

$${\Delta }_{0,0}(y;f)=\frac{-2f(y)}{y(1-y)},y\in (0,1).$$

$$cy(1-y)\le f(y).$$

$${\delta }_{n,k_n}^{+}(y_n;f)$$

$$\to \frac{f^{\prime }(x)}{1-y}-y\frac{f^{\prime }(x)}{1-y}=f^{\prime }(x)(n\to \infty ).$$

$$\delta^{+}_{n,k_{n}}(y_{n};f)=\frac{f\bigl{(}\frac{[r^{n}x]+1}{r^{n}}\bigr{)}-f(x)}{\frac{[r^{n}x]+1}{r^{n}}-x}\to f^{\prime}(x)\quad(n\to\infty).$$

$${\Delta }_{n,k}(y;f)\le -\frac{1}{t}.$$

$$q_f\left(t,x;\frac{k+y}{r^n}\right)=q_f\left(t,x;\frac{k}{r^n}\right).$$

$$x_1(n,k,y,t)=\frac{k}{r^n}+\frac{y}{2r^n}+t{\delta }_{n,k}^{-}(y;f).$$

$$\{\begin{array}{ll}{q}_f\left(t,x;\frac{k}{r^n}\right)\le q_f\left(t,x;\frac{k+y}{r^n}\right),& x\le x_1(n,k,y,t),\text{vskip}6.0ptplus2.0ptminus2.0pt\\ q_f\left(t,x;\frac{k}{r^n}\right)>q_f\left(t,x;\frac{k+y}{r^n}\right),& x_1(n,k,y,t)<x.\end{array}$$

$$q_f\left(t,x;\frac{k+y}{r^n}\right)=q_f\left(t,x;\frac{k+1}{r^n}\right)$$

$$x_2(n,k,y,t)=\frac{k}{r^n}+\frac{1+y}{2r^n}+t{\delta }_{n,k}^{+}(y;f).$$

$$\{\begin{array}{ll}{q}_f\left(t,x;\frac{k+y}{r^n}\right)\ge q_f\left(t,x;\frac{k+1}{r^n}\right),& x_2(n,k,y,t)\le x,\text{vskip}6.0ptplus2.0ptminus2.0pt\\ q_f\left(t,x;\frac{k+y}{2r^n}\right)<q_f\left(t,x;\frac{k+1}{r^n}\right),& x<x_2(n,k,y,t).\end{array}$$

$$x_1(n,k,y,t)\ge x_2(n,k,y,t).$$

$${\delta }_{n,k}^{-}(y;f)-{\delta }_{n,k}^{+}(y;f)\ge \frac{1}{2r^{n}t}.$$

$${\Delta }_{n,k}(y;f)\le -2c{r}^n\le -\frac{1}{t},$$

$${\Delta }_{n,k}(y;f)\le -\frac{1}{t}$$

$$U_{\psi }\left(\frac{k+y}{r^n}\right)-\sum _{j=0}^{n-1}\frac{1}{r^j}\psi \left(\frac{k+y}{r^{n-j}}\right)$$

$$=\sum _{j=n}^{\infty }\frac{1}{r^j}\psi \left(r^{j-n}(k+y)\right)=\sum _{j=n}^{\infty }\frac{1}{r^j}\psi \left(r^{j-n}y\right)=\frac{1}{r^n}{U}_{\psi }(y).$$

$$U_{\psi }\left(\frac{k}{r^n}\right)=\sum _{j=0}^{n-1}\frac{1}{r^j}\psi \left(\frac{k}{r^{n-j}}\right),U_{\psi }\left(\frac{k+1}{r^n}\right)=\sum _{j=0}^{n-1}\frac{1}{r^j}\psi \left(\frac{k+1}{r^{n-j}}\right).$$

Full text

The first author is supported in part by JSPS KAKENHI Nos. 15K04949 and 18K03360

The second author is supported in part by JSPS KAKENHI No. 16K17621

1. Introduction

Let us denote by $C_{p}(\mathbb{R})$ the set of all continuous and periodic functions $f:\mathbb{R}\to\mathbb{R}$ with period $1$ and $f(0)=0$. Throughout this paper, we assume that $r$ is an integer such that $r\geq 2$. Let $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$.

Our aim of this paper is to introduce and investigate the class $\mathcal{P}$ of functions in $C_{p}(\mathbb{R})$ defined as follows: Given a function $f\in C_{p}(\mathbb{R})$, we consider, for each $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$, the first-order forward and backward differences of $f$ at $\frac{k+y}{r^{n}}$ defined, respectively, by

[TABLE]

Definition 1.1**.**

Let $c>0$ be a given constant. A function $f\in C_{p}({\mathbb{R}})$ belongs to ${\mathcal{P}}_{c}$ if

[TABLE]

for all $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$. We use the notation $\mathcal{P}=\bigcup_{c>0}\mathcal{P}_{c}$. Note that both ${\mathcal{P}}_{c}$ and ${\mathcal{P}}$ depend on the choice of $r$ though we omit it in our notation.

Inequality (1.2) can be written equivalently as

[TABLE]

where $\Delta_{n,k}(y;f)$ is the second-order central difference defined by

[TABLE]

It is well-known that if a function $f:\mathbb{R}\to\mathbb{R}$ is concave and has the second derivative in some interval $I$, then $f^{\prime\prime}\leq 0$ in $I$. Even if $f$ is not twice differentiable, a discrete version of the estimate $\Delta_{n,k}(y,f)\leq 0$ still holds. Thus, the condition (1.3) can be regarded as a concavity-type estimate for $f$. Our definition of $\mathcal{P}$ requires a function to have the second-order differences which tend to $-\infty$ in the prescribed rate as $n\to\infty$.

Although Definition 1.1 seems to be independent of nowhere differentiable character, it turns out that each function in $\mathcal{P}$ is nowhere differentiable. This shows that our concavity-type estimate (1.3) is significantly different from a usual concavity since any concave function is twice differentiable almost everywhere.

We have two reasons to introduce and investigate the class $\mathcal{P}$. The first reason comes from a geometrical viewpoint. We show that each function in $\mathcal{P}$ has a geometrical characterization stated as follows: For any given function $f\in C_{p}({\mathbb{R}})$, let $\{q_{f}(t,x;z)\}_{z\in{\mathbb{R}}}$ be the family of parabolas defined by

[TABLE]

Then, we show that a function $f$ in $C_{p}({\mathbb{R}})$ belongs to $\mathcal{P}_{c}$ if and only if $f$ satisfies

• (F1)c

For all $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$ and $t\geq\frac{1}{2cr^{n}}$,

[TABLE]

Inequality (1.6) is a geometrical one related to position of the three parabolas. Another interpretation of (1.6) is that the function $q_{f}(t,x;\cdot)$ takes a minimum over the interval $[\frac{k}{r^{n}},\frac{k+1}{r^{n}}]$ at the endpoints.

The second reason comes from an analytic viewpoint. We consider the operator $U:C_{p}({\mathbb{R}})\ni\psi\mapsto U_{\psi}\in C_{p}({\mathbb{R}})$ defined by the series

[TABLE]

Such a series is known to generate nowhere differentiable functions under a suitable condition on $\psi$. We prove that the condition $U_{\psi}\in\mathcal{P}$ can be equivalently rephrased by the condition including the second-order differences of $\psi$. In fact, we establish

[TABLE]

whenever $\psi\in C_{p}(\mathbb{R})$ and $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$. When $n=0$, the first term of the right-hand side of (1.8) is interpreted as [math]. Thus, for a given $c>0$, we see that $U_{\psi}\in\mathcal{P}_{c}$ if and only if the right-hand side of (1.8) is less than or equal to $-2cr^{n}$ for all $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$. In other words, the class $\mathcal{P}$ is characterized via the operator $U$. Besides, making use of (1.8), we give some sufficient conditions on $\psi$ in order that $U_{\psi}\in\mathcal{P}$. We show that $U_{\psi}$ belongs to $\mathcal{P}$ if $\psi$ is concave on $[0,1]$. Also, even if $\psi$ is not concave on $[0,1]$, there is the case where $U_{\psi}$ belongs to $\mathcal{P}$ provided that $\psi$ is semiconcave on $[0,1]$ and satisfies some additional assumption. These simple sufficient conditions enable us to systematically construct a large number of examples of functions in the class $\mathcal{P}$ through the explicit formula (1.7).

A typical example of functions constructed by this procedure is the generalized Takagi function $\tau_{r}\in C_{p}(\mathbb{R})$ defined by

[TABLE]

where $d\in C_{p}({\mathbb{R}})$ is the distance function to the set $\mathbb{Z}$, that is,

[TABLE]

The celebrated Takagi function is given by $\tau_{2}$. The function $\tau_{2}$ is equivalent to the one first constructed by T. Takagi in 1903, who showed that $\tau_{2}$ is nowhere differentiable (see [17]). Its relevance in analysis, probability theory and number theory has been widely illustrated by many contributions, see for instance [17, 18, 1, 15]. Since $d$ is concave on $[0,1]$, we can show that $\tau_{r}$ belongs to $\mathcal{P}$ for any integer $r\geq 2$.

In connection to (F1)c, we also study the behavior of the Hamilton–Jacobi flow $\{H_{t}f\}_{t>0}$ starting from $f\in{\mathcal{P}}$, where

[TABLE]

This formula is widely used in the theory of viscosity solutions, and $H_{t}f$ is also referred to as an inf-convolution of $f$.

There are several papers related to our work. In [12], Hata and Yamaguti proposed a different generalization of the Tagaki function, the so-called Tagaki class, which includes not only nowhere differentiable functions, but also differentiable and even smooth ones. To analyze this class, they used some functional equations containing second-order central differences. Although we also use the second-order central difference $\Delta_{n,k}(y;f)$ of a function $f\in C_{p}(\mathbb{R})$, the frame and the purpose of the investigation of [12] are however rather different to ours. In [3, 13, 16], an inequality for approximate midconvexity of the Takagi function was investigated. A precise behavior of the flow $\{H_{t}\tau\}_{t>0}$ starting from the Takagi function is studied in [7].

The function $U_{\psi}$ of (1.7) has been considered by many authors. Cater [5] showed that if $\psi\in C_{p}({\mathbb{R}})$ is concave on the interval $[0,1]$ and $\psi$ takes its positive maximum over $[0,1]$ at $x=\frac{1}{2}$, then $U_{\psi}$ is nowhere differentiable. Although the connection between the concavity of $\psi$ and $U_{\psi}$ was already explored in [5], in this paper we show in addition that the formula (1.7) provides examples of functions in the class $\mathcal{P}$. Furthermore, we show that $U_{\psi}$ can belong to $\mathcal{P}$ even if $\psi\in C_{p}({\mathbb{R}})$ is not concave on $[0,1]$. Heurteaux [14] gave another sufficient conditions on $\psi\in C_{p}({\mathbb{R}})$ such that $U_{\psi}$ is nowhere differentiable. The set of maximum points in $[0,1]$ of the function $U_{\psi}$ was studied in [8] for $r=2$. However, all of the above papers neither characterize a class of nowhere differentiable functions nor introduce a class like $\mathcal{P}$.

The structure of the present paper is as follows. In Section 2 we prove nowhere differentiability and the geometrical characterization of a function in $\mathcal{P}$. Section 3 is devoted to the formula (1.8). We derive some sufficient conditions on $\psi\in\mathcal{P}$ in order that $U_{\psi}\in\mathcal{P}$. In Section 4, we study how the Hamilton-Jacobi flow $\{H_{t}f\}_{t>0}$ starting from $f\in{\mathcal{P}}$ behaves. Section 5 contains concluding remarks.

2. The class $\mathcal{P}$

In this section, we state and prove several results on the class $\mathcal{P}$. The first result of this section is Theorem 2.1, where we prove that each function in $\mathcal{P}$ is nowhere differentiable. The second result of this section is Theorem 2.3, which shows that a function $f$ in $C_{p}(\mathbb{R})$ belongs to $\mathcal{P}_{c}$ if and only if $f$ satisfies (F1)c.

Since we study periodic functions with period $1$, we often choose three points $\frac{k}{r^{n}}$, $\frac{k+y}{r^{n}}$, $\frac{k+1}{r^{n}}$ lying in $[0,1]$. For this reason, we prepare the set $\mathbb{A}$ of admissible triplets $(n,k,y)$ as

[TABLE]

For any $(n,k,y)\in\mathbb{A}$ we have $[\frac{k}{r^{n}},\frac{k+1}{r^{n}}]\subset[0,1]$. For a constant $c>0$, note that $f\in C_{p}({\mathbb{R}})$ belongs to $\mathcal{P}_{c}$ if and only if (1.2) is satisfied for all $(n,k,y)\in\mathbb{A}$.

We first derive a fundamental inequality for $f\in\mathcal{P}$. For $f\in C_{p}(\mathbb{R})$, we see by (1.4) that

[TABLE]

Thus, for $c>0$ and $y\in(0,1)$, we have $\Delta_{0,0}(y;f)\leq-2c$ if and only if

[TABLE]

Therefore we see that every $f\in\mathcal{P}_{c}$ satisfies (2.2) for any $y\in(0,1)$. In particular, when $f\in\mathcal{P}$, we have $f>0$ in $(0,1)$.

Now, we show that each function in $\mathcal{P}$ is nowhere differentiable. In what follows we write $[z]$ for $z\in\mathbb{R}$ to indicate the largest integer not exceeding $z$. We denote by $\mathbb{Q}_{r}$ the set of all rational numbers that can be written as $\frac{k}{r^{n}}$ for some $n\in\mathbb{N}$ and $k\in{\mathbb{Z}}$.

Theorem 2.1**.**

Each function in ${\mathcal{P}}$ is nowhere differentiable in $\mathbb{R}$.

Proof.

Fix $c>0$. Suppose that $f\in\mathcal{P}_{c}$ is differentiable at some point $x\in[0,1]$.

We set $k_{n}=[r^{n}x]$ for each $n\in\mathbb{N}$. Also, set $y_{n}=y$ if $x\in\mathbb{Q}_{r}$ and $y_{n}=r^{n}x-[r^{n}x]$ if $x\not\in\mathbb{Q}_{r}$, where $y\in(0,1)$ is an arbitrary constant. We claim that $\delta^{\pm}_{n,k_{n}}(y_{n};f)\to f^{\prime}(x)$ as $n\to\infty$. This gives a contradiction since taking the limit $n\to\infty$ in (1.2) along these $k_{n}$ and $y_{n}$ implies that $0\leq-c$.

When $x\in\mathbb{Q}_{r}$, we have $[r^{n}x]=r^{n}x$ for $n\in\mathbb{N}$ large. In fact, since $x\in\mathbb{Q}_{r}$, there are $n_{0}\in\mathbb{N}_{0}$ and $k_{0}\in\mathbb{Z}$ such that $x=\frac{k_{0}}{r^{n_{0}}}$, so that $r^{n}x=k_{0}r^{n-n_{0}}\in{\mathbb{N}}$ if $n\geq n_{0}$. For $n\geq n_{0}$ we find that

[TABLE]

In the same manner, we deduce that $\delta^{-}_{n,k_{n}}(y_{n};f)\to f^{\prime}(x)$ as $n\to\infty$.

Next, let $x\not\in{\mathbb{Q}}_{r}$. We then have $[r^{n}x]<r^{n}x<[r^{n}x]+1$ for each $n\in\mathbb{N}$. This implies that $y_{n}\in(0,1)$ for each $n\in\mathbb{N}$ and that $\frac{[r^{n}x]}{r^{n}}\to x$ as $n\to\infty$. Thus,

[TABLE]

Similarly, it follows that $\delta^{-}_{n,k_{n}}(y_{n};f)\to f^{\prime}(x)$. This completes the proof. ∎

Next, we show that a function $f$ in $C_{p}({\mathbb{R}})$ belongs to $\mathcal{P}_{c}$ if and only if $f$ satisfies (F1)c. To prove this, the following proposition is essential:

Proposition 2.2**.**

Let $(n,k,y)\in\mathbb{A}$ and $t\in(0,\infty)$. Then, for any $f\in C_{p}({\mathbb{R}})$, inequality (1.6) holds if and only if

[TABLE]

Proof.

Fix $(n,k,y)\in\mathbb{A}$ and $t\in(0,\infty)$. Let $x_{1}(n,k,y,t)$ be the unique solution of the equation

[TABLE]

By direct calculation,

[TABLE]

Then, we have

[TABLE]

Similarly, the unique solution $x_{2}(n,k,y,t)$ of the equation

[TABLE]

is given by

[TABLE]

Furthermore,

[TABLE]

Then, a geometrical investigation shows that inequality (1.6) holds if and only if

[TABLE]

By (2.4) and (2.5), we see that inequality (2.6) holds if and only if

[TABLE]

The desired inequality follows immediately from (1.4). ∎

Now, we state the second result of this section.

Theorem 2.3**.**

Let $f\in C_{p}(\mathbb{R})$ and let $c>0$ be a constant. Then, $f$ satisfies (F1)c if and only if $f\in\mathcal{P}_{c}$.

Proof.

Assume first that $f\in\mathcal{P}_{c}$. Fix $(n,k,y)\in\mathbb{A}$ and $t\geq\frac{1}{2cr^{n}}$ arbitrarily. By (1.3) and (1.4), we have

[TABLE]

and so (1.6) holds by Proposition 2.2. Thus we see that $f$ satisfies (F1)c.

Next, assume that (F1)c holds. Then, by Proposition 2.2, we see that

[TABLE]

for all $(n,k,y)\in\mathbb{A}$ and $t\geq\frac{1}{2cr^{n}}$. Letting $t=\frac{1}{2cr^{n}}$, we conclude that $f\in\mathcal{P}_{c}$. ∎

3. Functions $U_{\psi}$ and $\mathcal{P}$

In this section, we give sufficient conditions on $\psi\in C_{p}(\mathbb{R})$ in order that $U_{\psi}\in\mathcal{P}$, where $U$ is the operator defined by (1.7). The results enable us to generate a large number of functions in $\mathcal{P}$ through the explicit formula (1.7). We also give some examples of $\psi\in C_{p}(\mathbb{R})$ for which $U_{\psi}\not\in\mathcal{P}$.

The following theorem provides a representation of $\Delta_{n,k}(U_{\psi};y)$ in terms of $\Delta_{n,k}(\psi;y)$, which plays a crucial role to study if $U_{\psi}\in\mathcal{P}$. Note that, for every $\psi\in C_{p}(\mathbb{R})$, we have $U_{\psi}\in C_{p}(\mathbb{R})$ and $U_{\psi}(0)=0$ by the definition of $U_{\psi}$.

Theorem 3.1**.**

Let $\psi\in C_{p}(\mathbb{R})$. Then, (1.8) holds for each $(n,k,y)\in\mathbb{A}$. When $n=0$, the first term of the right-hand side of (1.8) is interpreted as [math].

Proof.

Let $(n,k,y)\in\mathbb{A}$. When $n=0$, we have $k=0$, so that (1.8) follows from (2.1) since $U_{\psi}(0)=0$. If $n\geq 1$, then

[TABLE]

This is valid even for $y=0$ and $y=1$. Since $U_{\psi}(0)=U_{\psi}(1)=0$, we have

[TABLE]

We therefore have

[TABLE]

This implies (1.8). ∎

Applying Theorem 3.1, we derive some sufficient conditions on $\psi\in C_{p}(\mathbb{R})$ that guarantee $U_{\psi}\in\mathcal{P}$. As a typical result, it turns out that $U_{\psi}\in\mathcal{P}$ if $\psi$ is concave in $[0,1]$ and positive in $(0,1)$.

Let us recall a notion of concavity. A function $g:[0,1]\to\mathbb{R}$ is said to be concave on $[0,1]$ if the inequality

[TABLE]

holds for all $x,y\in[0,1]$ and $\lambda\in[0,1]$. If the reversed inequality holds, then $g$ is said to be convex. For a constant $\alpha\geq 0$, a function $g$ on $[0,1]$ is said to be $\alpha$-semiconcave on $[0,1]$ if $g(x)+\frac{\alpha}{2}x(1-x)$ is concave on $[0,1]$. This is equivalent to the condition that $g(x)-\frac{\alpha}{2}x^{2}$ is concave on $[0,1]$.

Remark 3.2*.*

1. (i)

Let $\psi\in C_{p}(\mathbb{R})$ and assume that $\psi$ is concave on some interval $I$. Then it is easy to see that $\Delta_{n,k}(y;\psi)\leq 0$ for all $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$ such that $[\frac{k}{r^{n}},\frac{k+1}{r^{n}}]\subset I$. More generally, if $\psi\in C_{p}({\mathbb{R}})$ is $\alpha$-semiconcave on $I$, then we have $\Delta_{n,k}(y;\psi)\leq\alpha$ for all $(n,k,y)\in\mathbb{N}_{0}\times\mathbb{Z}\times(0,1)$ such that $[\frac{k}{r^{n}},\frac{k+1}{r^{n}}]\subset I$. The reversed inequalities hold for ($\alpha$-semi)convex functions. 2. (ii)

If $\psi\in C_{p}({\mathbb{R}})$ is concave on $[0,1]$, then we have $\Delta_{n,k}(y,\psi)\leq 0$ for all $(n,k,y)\in\mathbb{A}$ by (i). However, the converse is not true in general: that is, even if $\Delta_{n,k}(y,\psi)\leq 0$ for all $(n,k,y)\in\mathbb{A}$, we cannot say that $\psi$ is concave on $[0,1]$. Every $f\in{\mathcal{P}}$ gives a counterexample to this. In fact, $\Delta_{n,k}(y,f)\leq 0$ for all $(n,k,y)\in\mathbb{A}$, but $f$ is never concave on $[0,1]$ by Theorem 2.1, since a concave function must be differentiable almost everywhere.

We first prepare inequalities involving $U_{\psi}$ and the generalized Takagi function $\tau_{r}$ defined in (1.9). Recall that $d$ is the distance function given by (1.10).

Lemma 3.3**.**

Let $\psi\in C_{p}(\mathbb{R})$. Assume that there exists a constant $m>0$ such that $md(x)\leq\psi(x)$ for all $x\in[0,1]$. Then, we have

[TABLE]

Proof.

It follows from our assumption that $md(r^{j}x)\leq\psi(r^{j}x)$ for all $x\in[0,1]$ and $j\in\mathbb{N}_{0}$. Thus, $m\tau_{r}(x)\leq U_{\psi}(x)$ by taking the sum.

It remains to prove that

[TABLE]

Let

[TABLE]

Since $F\leq\tau_{r}$, it suffices to show that $G(x)\leq F(x)$ for $x\in[0,1]$. As $F$ and $G$ are symmetric about $x=\frac{1}{2}$, we may assume that $x\in[0,\frac{1}{2}]$. Note that

[TABLE]

When $0\leq x\leq\frac{1}{r}$, we have

[TABLE]

Thus $G(x)\leq F(x)$. Next, let $\frac{1}{r}\leq x\leq\frac{1}{2}$. Then,

[TABLE]

Hence, we conclude (3.2). ∎

Remark 3.4*.*

Assume that $\psi\in C_{p}(\mathbb{R})$ is concave in $[0,1]$ and $\psi>0$ in $(0,1)$. Then, we have

[TABLE]

and thus $\psi$ satisfies the assumption in Lemma 3.3 for $m=2\psi(\frac{1}{2})$. Indeed, by the concavity of $\psi$, its graph lies above the segment connecting $(0,\psi(0))$ and $(\frac{1}{2},\psi(\frac{1}{2}))$ and the segment connecting $(\frac{1}{2},\psi(\frac{1}{2}))$ and $(1,\psi(1))$. This shows (3.3) since $\psi(0)=\psi(1)=0$.

Now, we state the main result of this section.

Theorem 3.5**.**

Let $\psi\in C_{p}(\mathbb{R})$. Assume that there exist two constants $m>0$ and $\alpha\geq 0$ such that

1. (i)

$md(x)\leq\psi(x)$* for all $x\in[0,1]$.* 2. (ii)

$\Delta_{n,k}(y;\psi)\leq\alpha$* for all $(n,k,y)\in\mathbb{A}$.*

If $2mr>\alpha$, then $U_{\psi}\in\mathcal{P}_{c}$ with $c=\frac{2mr-\alpha}{2(r-1)}$.

Proof.

Let us derive $\Delta_{n,k}(y;U_{\psi})\leq-2cr^{n}$ for a fixed $(n,k,y)\in\mathbb{A}$. From Lemma 3.3 it follows that

[TABLE]

If $n=0$, we see by (2.1) that $\Delta_{0,0}(y;U_{\theta})\leq-\frac{2mr}{r-1}<-2c$. For $n\geq 1$ we have

[TABLE]

Thus, by (1.8)

[TABLE]

which proves the theorem. ∎

Let us denote by $E$ the set of $\psi\in C_{p}(\mathbb{R})$ satisfying (i) and (ii) in Theorem 3.5 for some $m>0$ and $\alpha\geq 0$ with $2mr>\alpha$. Theorem 3.5 asserts that $U_{\psi}\in\mathcal{P}$ for every $\psi\in E$. We give typical classes that are included in $E$.

Proposition 3.6**.**

The set $E$ includes the following two sets:

• $(1)$

\mathit{SC}_{0}:=\{\psi\in C_{p}(\mathbb{R})\mid\mbox{\psi$is concave in$[0,1]$and$\psi>0$in$(0,1)}\}.

• $(2)$

$\mathcal{P}$.

Proof.

(1) Let $\psi\in\mathit{SC}_{0}$. It follows from Remark 3.4 that $\psi$ satisfies Theorem 3.5-(i) for $m=2\psi(\frac{1}{2})$, while we can take $\alpha=0$ in Theorem 3.5-(ii) by Remark 3.2-(i). Since $2mr>\alpha$, we have $\psi\in E$ and $U_{\psi}\in\mathcal{P}_{c}$ with $c=\frac{2r}{r-1}\psi(\frac{1}{2})$.

(2) Let $\psi\in\mathcal{P}_{c}$ for some $c>0$. By (2.2), we can take $m=c$ in Theorem 3.5-(i). We also take $\alpha=0$ in Theorem 3.5-(ii) by the definition of $\mathcal{P}_{c}$. Since $2mr>\alpha$, we conclude that $\psi\in E$ and $U_{\psi}\in\mathcal{P}_{c^{\prime}}$ with $c^{\prime}=\frac{cr}{r-1}$. ∎

Note that the two sets $\mathit{SC}_{0}$ and $\mathcal{P}$ above are mutually disjoint, since a concave function is differentiable almost everywhere. Also, if $\psi$ belongs to $\mathcal{P}$, then $U_{\psi}$ also belongs to $\mathcal{P}$ since $\mathcal{P}\subset E$ by Proposition 3.6-(2). Thus, $\mathcal{P}$ is an invariant set under the operator $U$.

Remark 3.7*.*

By Proposition 3.6-(1) and its proof, we see that the generalized Takagi function $\tau_{r}$ belongs to $\mathcal{P}_{c}$ with $c=\frac{r}{r-1}$ since $d\in C_{p}({\mathbb{R}})$ is concave in $[0,1]$ and $d(\frac{1}{2})=\frac{1}{2}$. In particular, the Takagi function $\tau_{2}$ is in $\mathcal{P}_{2}$ for $r=2$.

If $\psi\in C_{p}(\mathbb{R})$ is $\alpha$-semiconcave in $[0,1]$, then (ii) in Theorem 3.5 is fulfilled by Remark 3.2-(i). However, (i) does not hold in general even if $\psi>0$ in $(0,1)$. One may then wonder if $U_{\psi}$ belongs to $\mathcal{P}$ for $\psi$ in

[TABLE]

with $\alpha>0$. The answer is no. Besides, $U_{\psi}$ for $\psi\in\mathit{SC}_{\alpha}$ does not necessarily possess nowhere differentiable character. Namely, for every $\alpha>0$ there are the following three examples of $\psi\in\mathit{SC}_{\alpha}$:

• (A)

$U_{\psi}\in\mathcal{P}$ and $\psi\not\in\mathit{SC}_{0}$.

• (B)

$U_{\psi}\not\in\mathcal{P}$ and $U_{\psi}$ is nowhere differentiable in $[0,1]$.

• (C)

$U_{\psi}\not\in\mathcal{P}$ and $U_{\psi}\in C^{\infty}((0,1))$.

Let us give an example of $\psi\in\mathit{SC}_{\alpha}$ satisfying each (A)–(C).

Example 3.8**.**

For constants $a,b>0$, let $\psi_{0}=ad+bd^{2}\in C_{p}(\mathbb{R})$. Then, $\psi_{0}$ is not concave on $[0,1]$ but $2b$-semiconcave on $[0,1]$. In addition, when $ar>b$, $U_{\psi_{0}}\in\mathcal{P}$. We thus obtain a function satisfying (A).

Indeed, since $\psi_{0}(x)=ax+bx^{2}$ on $[0,\frac{1}{2}]$, $\psi_{0}$ is not concave on $[0,1]$. Also, we have $\psi_{0}(x)+bx(1-x)=(a+b)d(x)$ on $[0,1]$, and so $\psi_{0}$ is $2b$-semiconcave on $[0,1]$. Finally, since $\psi_{0}\geq ad$ on $[0,1]$, we can take $m=a$ and $\alpha=2b$ in Theorem 3.5. Thus, $\psi_{0}\in E$ and so $U_{\psi_{0}}\in\mathcal{P}$.

This example also shows that $\mathit{SC}_{0}\cup\mathcal{P}\subsetneq E$.

Let us next discuss the example of (B). Let $\theta\in C_{p}(\mathbb{R})$ be a function such that

[TABLE]

We now apply [14, Theorem 3.1], which asserts that, if $\psi\in C_{p}(\mathbb{R})\cap C^{1}(\mathbb{R})$ and $\psi^{\prime}$ is Hölder continuous in $\mathbb{R}$, then $U_{\psi}$ is nowhere differentiable in $\mathbb{R}$. Since $\theta$ satisfies these conditions, we deduce that $U_{\theta}$ is nowhere differentiable in $\mathbb{R}$. However, $U_{\theta}$ does not belong to $\mathcal{P}$ as shown below.

Theorem 3.9**.**

$\Delta_{n,0}(\frac{1}{r};U_{\theta})=-\frac{2}{r-1}$* for each $n\in\mathbb{N}_{0}$. Thus, $U_{\theta}\not\in\mathcal{P}$.*

Proof.

Let $n\in\mathbb{N}_{0}$. We have

[TABLE]

Thus,

[TABLE]

When $n=0$, this and (2.1) shows that $\Delta_{0,0}(\frac{1}{r};U_{\theta})=-\frac{2}{r-1}$. Let $n\geq 1$. Since \Delta_{m,0}\bigl{(}\frac{1}{r},{\theta})=2 for any $m\in\mathbb{N}$, it follows from Theorem 3.1 that

[TABLE]

The proof is complete. ∎

Let $\alpha>0$. Since $\theta\in C^{2}(\mathbb{R})$, we have $\varepsilon\theta\in\mathit{SC}_{\alpha}$ if $\varepsilon>0$ is sufficiently small. Also, it is easy to see that $U_{\varepsilon\theta}$ is still nowhere differentiable and $U_{\varepsilon\theta}\not\in\mathcal{P}$. We thus obtain a function satisfying (B).

Example 3.10**.**

Let us give an example of a function satisfying (C). Define

[TABLE]

Then, by the definition of $U_{\psi}$, we easily see that $U_{\psi}(x)=|\sin(\pi x)|\in C_{p}(\mathbb{R})$. Thus $U_{\psi}\in C^{\infty}((0,1))$ and in particular $U_{\psi}\not\in\mathcal{P}$ as required in (C).

Let us next check that $\psi\in\mathit{SC}_{\alpha}$ for some $\alpha>0$. The positivity of $\psi$ in $(0,1)$ follows from straightforward calculation, and so we omit the proof. Next, since functions $\frac{1}{r}\sin(\pi rx)$ and $-\frac{1}{r}\sin(\pi rx)$ are semiconcave, the minimum of them $-\frac{1}{r}|\sin(\pi rx)|$ is also semiconcave. Therefore, $\psi$ being the sum of two semiconcave functions in $[0,1]$ is semiconcave in $[0,1]$.

Similarly to the previous example, for a given $\alpha>0$, we have $\varepsilon\psi\in\mathit{SC}_{\alpha}$ if $\varepsilon>0$ is sufficiently small. A function satisfying (C) has thus been obtained.

We conclude this section by studying if a Weierstrass type function belongs $\mathcal{P}$.

Example 3.11**.**

The famous Weierstrass function $W$ is given by

[TABLE]

where $a\in(0,1)$ and $b$ is an odd integer with $ab>1+\frac{3\pi}{2}$. Note that $\rho$ is continuous and periodic on $\mathbb{R}$ with period $2$ and $\rho(0)\neq 0$. Since we consider functions $\psi$ in $C_{p}(\mathbb{R})$ with $\psi(0)=0$ in this paper, we study $U_{\eta}$ for $\eta(x)=\sin(2\pi x)\in C_{p}(\mathbb{R})$ instead of $W$. By Hardy [11], it is shown that $U_{\eta}$ is nowhere differentiable. We also remark that $\eta$ possesses a balance of convexity and concavity properties, since it is concave on $[0,\frac{1}{2}]$ and convex on $[\frac{1}{2},1]$.

We claim that $U_{\eta}$ does not belong to $\mathcal{P}$. In fact, noting that $\eta(\frac{r^{j}}{2})=\sin(\pi r^{j})=0$ for all $j\in\mathbb{N}_{0}$, we see that $U_{\eta}(\frac{1}{2})=0$ by the definition of $U_{\eta}$. This implies that $U_{\eta}\not\in\mathcal{P}$ since, if $U_{\eta}\in\mathcal{P}$, we have $U_{\eta}>0$ in $(0,1)$ by (2.2).

4. The behavior of $\{H_{t}f\}_{t>0}$ for $f\in\mathcal{P}$

In this section we consider the behavior of the Hamilton-Jacobi flow $\{H_{t}f\}_{t>0}$ for $f\in\mathcal{P}$, where $H_{t}f$ is the function defined by (1.11). It is known that $H_{t}f$ belongs to $C_{p}({\mathbb{R}})$ and uniformly approximates $f$ as $t$ goes to [math] (see [4, Chapter 3.5]). Also, $H_{t}f$ is a unique viscosity solution of the initial value problem of the Hamilton–Jacobi equation:

[TABLE]

(cf. [6]). Here, $u_{t}(t,x)=\frac{\partial u}{\partial t}(t,x)$ and $u_{x}(t,x)=\frac{\partial u}{\partial x}(t,x)$.

First of all, we prove that the range of $z$ in (1.11) can be reduced.

Lemma 4.1**.**

Let $f\in C_{p}(\mathbb{R})$. If $f(z)\geq 0$ for all $z\in[0,1]$, then

[TABLE]

Proof.

Fix $(t,x)\in(0,\infty)\times[0,1]$. We first let $z<0$. Since $f(z)\geq 0$, the geometrical investigation implies that $q_{f}(t,x;z)>q_{f}(t,x;0)$. Thus, the minimum in (1.11) is never attained for $z<0$. The same arguments show that $z>1$ is not a minimizer of (1.11), and hence (4.2) holds. ∎

Now, we state the main result of this section.

Theorem 4.2**.**

Let $f\in{\mathcal{P}}_{c}$ for $c>0$. Then, the following holds:

• (F2)c

For all $n\in\mathbb{N}_{0}$,

[TABLE]

Proof.

This is a consequence of (4.2) and (F1)c. In fact, since $f\in{\mathcal{P}}_{c}$ satisfies the inequality $f(z)\geq 0$ for $z\in[0,1]$ by (2.2), we have (4.2), while Theorem 2.3 guarantees that (F1)c holds. ∎

By Theorem 4.2 we see that $H_{t}f$ with $f\in\mathcal{P}_{c}$ is a piecewise quadratic function in $[0,1]$ for all $t>0$ and that the $x$-coordinate of each vertex of the parabolas making up $H_{t}f$ always belongs to $\mathbb{Q}_{r}$. In general it is known that $H_{t}f$ for $f\in C_{p}(\mathbb{R})$ is $\frac{1}{2t}$-semiconcave in $[0,1]$ for all $t>0$. For $f\in\mathcal{P}_{c}$ we deduce from (4.3) that

[TABLE]

for $(t,x)\in[\frac{1}{2cr^{n}},\infty)\times[0,1]$. This shows that $H_{t}f(x)-\frac{x^{2}}{2t}$ is not only concave but also piecewise linear in $[0,1]$.

One may ask if, conversely, a function $f\in C_{p}(\mathbb{R})$ satisfying (F2)c for some $c>0$ is nowhere differentiable. We have no complete answer to this question at the moment. However, we can prove that such an $f$ is non-differentiable on a dense subset of $\mathbb{R}$. In general this is not enough to infer that it is nowhere differentiable, as is shown by the Riemann function. Indeed, let $R$ be the Riemann function defined by

[TABLE]

Set

[TABLE]

By Hardy [11] and Gerver [9, 10], it is shown that $R$ is differentiable on the set $F$ and that $R$ is non-differetiable on the set $(\mathbb{R}\setminus\mathbb{Q})\cup(\mathbb{Q}\setminus F)$.

Theorem 4.3**.**

Let $f\in C_{p}(\mathbb{R})$ and let $c>0$ be a constant. Assume that (F2)c holds. Then, there exists a dense subset of the interval $[0,1]$ such that $f$ is non-differentiable at each point of this subset.

We denote by $D^{-}f(x)$ the subdifferential of $f$ at $x$, that is, the set of $\phi^{\prime}(x)$ such that $\phi\in C^{1}$ near $x$ and $f-\phi$ has a local minimum at $x$. We list basic properties of the subdifferential used in the proof of Theorem 4.3. Let $f\in C_{p}(\mathbb{R})$ and $x\in\mathbb{R}$.

1. (I)

If $f$ is differentiable at $x$, then $D^{-}f(x)=\{f^{\prime}(x)\}$ ([2, Lemma II.1.8-(b)]); 2. (II)

Let $t>0$ and choose $z\in\mathbb{R}$ such that $H_{t}f(x)=q_{f}(t,x;z)$. Then $\frac{x-z}{t}\in D^{-}f(z)$ ([2, Lemma II.4.12-(iii)]).

Proof of Theorem 4.3.

Fix $x_{0}\in(0,1)$ and $\varepsilon>0$, and let $I=(x_{0}-\varepsilon,x_{0}+\varepsilon)$. We prove that there is some $z\in I$ such that $f$ is not differentiable at $z$. We may assume that $\varepsilon<\min\{x_{0},\,1-x_{0}\}$, so that $I\subset[0,1]$. Let $t\in(0,\frac{\varepsilon^{2}}{2M})$, with $M>0$ the oscillation of $f$, that is, $M=\sup_{\mathbb{R}}f-\inf_{\mathbb{R}}f$. Since $H_{t}f$ is represented by (4.3) with $n$ such that $t\geq\frac{1}{2cr^{n}}$, there exists some $\delta\in(0,\varepsilon)$ such that $H_{t}f=q_{f}(t,\cdot;z)$ in $J:=[x_{0}-\delta,x_{0}]\subset I$ with $z=\frac{k}{r^{n}}$ for some $k\in\{0,1,2,3,\dots,r^{n}\}$. The choice of $t$ then guarantees that $z\in I$. Indeed, we have

[TABLE]

and hence $(x_{0}-z)^{2}\leq 2t(f(x_{0})-f(z))\leq 2Mt<\varepsilon^{2}$, that is, $z\in I$.

It follows from (II) that $\frac{x-z}{t}\in D^{-}f(z)$ for all $x\in J$. This implies that $[\frac{x_{0}-\delta-z}{t},\frac{x_{0}-z}{t}]\subset D^{-}f(z)$: that is, $D^{-}f(z)$ is not a singleton. Hence we conclude by (I) that $f$ is not differentiable at $z$. ∎

Remark 4.4*.*

The above proof actually shows that the dense set we found is a subset of $\mathbb{Q}_{r}$.

5. Concluding remark

We conclude this paper by mentioning another possible definition of $\mathcal{P}_{c}$. Let us define $\mathcal{P}^{\prime}_{c}$ as the set of all $f\in C_{p}(\mathbb{R})$ such that there exists an infinite subset $\mathbb{N}^{\prime}\subset\mathbb{N}_{0}$ such that $f$ satisfies (1.2) for all $(n,k,y)\in\mathbb{A}$ with $n\in\mathbb{N}^{\prime}$. In other words, we require (1.2) only for some subsequence of $n\in\mathbb{N}_{0}$. Even if this generalized class $\mathcal{P}^{\prime}_{c}$ is used, one can easily see that Theorem 2.3 is obtained in a suitable sense. Namely, $f\in\mathcal{P}^{\prime}_{c}$ if and only if $f$ satisfies (F1)c with “For all $n\in\mathbb{N}^{\prime}$” instead of “For all $n\in\mathbb{N}_{0}$”. The proof is almost the same as before.

Moreover, Theorem 2.1 is true for a function in $\mathcal{P}^{\prime}:=\bigcup_{c>0}\mathcal{P}^{\prime}_{c}$ since the proof still works when taking the limit along $\mathbb{N}^{\prime}$. The formula (1.7) still gives many examples of functions in $\mathcal{P}^{\prime}$. Though $\mathcal{P}^{\prime}$ provides a more general class than does $\mathcal{P}$, there are, however, no essential changes or difficulties in the proofs. For this reason, for simplicity of presentation, the authors decided to give results in this paper for $\mathcal{P}_{c}$ instead of $\mathcal{P}^{\prime}_{c}$.

Acknowledgement

Antonio Siconolfi appreciates funding for selected research from the Faculty of Science, University of Toyama. It enabled him to visit the University of Toyama in March, 2018.