Almost Periodic Functions with Values in a Non-locally Convex Space

Abstract

In this section, (X, +, ⋅, ||⋅||) will be a p-Fréchet space with 0 < p < 1 (over the field $\Phi =\mathbb {R}$ or $\mathbb {C}$). Also, denote D(x, y) = ||x − y||.

Similarly to, p. 137, a trigonometric polynomial of degree ≤ n with coefficients (and values) in the p-Fréchet space X is defined as a finite sum of the form $T_{n}(t)=\displaystyle \sum \limits _{k=1}^{n}c_{k}e^{i\lambda _{n}t}$, where c _k ∈ X, k = 1, …, n.

In this section, (X, +, ⋅, ||⋅||) will be a p-Fréchet space with 0 < p < 1 (over the field $\Phi =\mathbb {R}$ or $\mathbb {C}$). Also, denote D(x, y) = ||x − y||.

Similarly to [22], p. 137, a trigonometric polynomial of degree ≤ n with coefficients (and values) in the p-Fréchet space X, is defined as a finite sum of the form $T_{n}(t)=\displaystyle \sum \limits _{k=1}^{n}c_{k}e^{i\lambda _{n}t}$, where c _k ∈ X, k = 1, …, n.

Also, recall that $f:\mathbb {R}\to X$ is said to be continuous on $x_{0}\in \mathbb {R}$ if ∀𝜖 > 0, there exists δ > 0 such that $||f\left ( x\right ) - f\left ( x_{0}\right )|| <\epsilon$, whenever $x\in \mathbb {R}$, $\left | x-x_{0}\right | <\delta$. From the triangle inequality satisfied by the p-norm ||⋅||, it easily follows the inequality | ||x||−||y|| |≤||x − y||, which immediately implies that if f is continuous in x ₀ as above, then the real-valued function ||f(t)|| is also continuous at x ₀.

1 Definitions and Properties

In this section, starting from a Bohr-kind definition for the almost periodicity, we develop a theory of almost periodic functions with values in a p-Fréchet space, 0 < p < 1, similar to that for functions with values in a Banach space.

The following three points in Definition 3.1 represent the basic concepts in the theory of almost periodic functions with values in the p-Fréchet space X.

Definition 9.1

Let $f:\mathbb {R}\to X$ be continuous on $\mathbb {R}$.

• (i) We say that f is almost periodic if ∀𝜖 > 0, there exists $l\left ( \epsilon \right ) >0$ such that any interval of length $l\left ( \epsilon \right )$ of the real line contains at least one point ξ with

  $$\displaystyle \begin{aligned}||f\left( t+\xi\right) - f\left(t\right)|| <\epsilon,\forall t\in\mathbb{R}.\end{aligned}$$

• (ii) We say that f is normal if for any sequence $F_{n}:\mathbb {R} \to X$ of the form $F_{n}\left ( x\right ) =f\left ( x+h_{n}\right )$, $n\in \mathbb {N}$, where $\left ( h_{n}\right ) _{n}$ is a sequence of real numbers, one can extract a subsequence of $\left ( F_{n}\right ) _{n}$, converging uniformly on $\mathbb {R}$ (i.e. $\forall \left ( h_{n}\right ) _{n}$, $\exists \left ( F_{n_{k}}\right )$, $\exists F:\mathbb {R}\to X$ (which may depend on $\left ( h_{n}\right ) _{n}$), such that $\displaystyle \lim _{k\to \infty }||F_{n_{k}}\left ( x\right ) - F\left (x\right )||=0$, uniformly with respect to $x\in \mathbb {R}$).

• (iii) We say that f has the approximation property, if ∀𝜖 > 0, there exists some trigonometric polynomial T with coefficients in X, such that $||f\left ( x\right ) - T\left ( x\right )|| <\epsilon$, $\forall x\in \mathbb {R}$.

Let us denote $AP\left ( X\right ) =\left \{ f:\mathbb {R}\rightarrow X;f\mbox{ is almost periodic}\right \}$. The next two theorems show that $AP\left ( X\right )$ is a subclass of uniformly continuous bounded functions.

Remark 9.2

We can reformulate 9.1(i), as follows: $f:\mathbb {R}\to X$ is called almost periodic if for every ε > 0, there exists a relatively dense set {τ}_ε, such that

$$\displaystyle \begin{aligned}\sup_{t\in \mathbb{R}}||f(t+\tau)-f(t)||\le \varepsilon, \mbox{ for all } \tau \in\{\tau\}_{\varepsilon}.\end{aligned}$$

Also, each τ ∈{τ}_ε is called ε-almost period of f.

Remark 9.3

Theorems 10.5, 8.5, 8.6, and 8.10 and Remark 8.8 remain valid in p-Fréchet spaces, 0 < p < 1.

Theorem 9.4

If f has the approximation property, then it is almost periodic.

Proof

A function $f:\mathbb R\to \mathbb X$ is said to be τ-periodic if f(t + τ) = f(t) for all $t\in \mathbb R$. Obviously, any trigonometric polynomial with values in $\mathbb X$ is almost periodic. This with Theorem 8.6 completes the proof. □

Remark 9.5

Let us denote $AP\left ( X\right ) =\left \{ f:\mathbb {R}\rightarrow X;f\mbox{ is B-almost periodic}\right \},$ and for $f\in AP\left ( X\right )$, let us define $\left \| f\right \|{ }_{b} =\sup \left \{ \left \| f\left ( t\right ) \right \|;t\in \mathbb {R}\right \}$. By Theorem 3.2, we get $\left \| f\right \|{ }_{b} <+\infty$. It easily follows that ||⋅||_b is also a p-norm on the space

$$\displaystyle \begin{aligned}BC\left(\mathbb{R}, X\right) =\{f:\mathbb{R}\rightarrow X \mbox{ is continuous and bounded on } \mathbb{R}\}.\end{aligned}$$

In addition, since (X, D), where D is defined by D(x, y) = ∥x − y∥, is a complete metric space, by standard reasonings, it follows that $BC(\mathbb {R}, X)$ becomes complete metric space with respect to the metric D _b(f, g) = ||f − g||_b, that is, $(BC(\mathbb {R}, X), ||\cdot ||{ }_{b})$ becomes a p-Fréchet space.

Then, Theorems 3.2 and 3.5 show that $AP\left ( X\right )$ is a closed subset of $BC\left (\mathbb {R}, X\right )$, i.e. $\left ( AP\left ( X \right ) ,D_{b}\right )$ is a complete metric space, and therefore (AP(X), ||⋅||_b) becomes a p-Fréchet space. By similar reasonings with those in the proofs of Theorems 6.9 and 6.10 in [22], pp. 142–143 (where we define on X ^m the p-norm $||x||{ }_{m}=\displaystyle \sum \limits _{k=1}^{m}||x_{k}||$ and the metric $D_{m}\left ( x,y\right ) =\displaystyle \sum \limits _{i=1}^{m}D\left ( x_{i},y_{i}\right )$, $\forall x=\left ( x_{1},\ldots ,x_{m}\right )$, $y=\left ( y_{1},\ldots ,y_{m}\right ) \in X^{m}$), we can state the following compactness criterion.

Theorem 9.6

The necessary and sufficient condition that a family $\mathcal {A}\subset AP( \mathbb X )$ be relatively compact is that the following properties hold true:

• (i) $\mathcal {A}$ is equicontinuous;

• (ii) $\mathcal {A}$ is equi-almost periodic;

• (iii) for any $t\in \mathbb {R}$ , the set of values of functions from $\mathcal {A}$ be relatively compact in $\mathbb X$.

In what follows, we consider the concept of Bochner’s transform. Thus, Bochner’s transform of f in $BC(\mathbb {R}, X)$ is denoted by $B(f):=\tilde {f}$ and is defined by $\tilde {f}:\mathbb {R}\to BC(\mathbb {R}, X)$, $\tilde {f}(s)\in BC(\mathbb {R}, X)$, $\tilde {f}(s)(t)=f(t+s)$, for all $t\in \mathbb {R}$.

The properties of the Bochner transform can be summarized in the following theorem:

Theorem 9.7

• (i) $||\tilde {f}(s)||{ }_{b}=||f(\cdot +s)||{ }_{b}=||\tilde {f}(0)||{ }_{b}, \mathit{\mbox{ for all }} s\in \mathbb {R};$

• (ii) $||\tilde {f}(s+\tau )-\tilde {f}(s)||{ }_{b}=sup\{||f(t+\tau )-f(t)|| ; t\in \mathbb {R}\} =||\tilde {f}(\tau )-\tilde {f}(0)||{ }_{b}, \mathit{\mbox{ for all }} s, \tau \in \mathbb {R};$

• (iii) f is B-almost periodic if and only if $\tilde {f}$ is B-almost periodic, with the same set of ε-almost periods {τ}_ε;

• (iv) $\tilde {f}$ is B-almost periodic, if and only if there exists a relatively dense sequence in $\mathbb {R}$ , denoted by $\{s_{n}; n\in \mathbb {N}\}$ , such that the set of functions $\{\tilde {f}(s_{n}) ; n\in \mathbb {N}\}$ is relatively compact in the complete metric space $(BC(\mathbb {R}, \mathbb X), D_{b})$;

• (v) $\tilde {f}$ is B-almost periodic, if and only $\tilde {f}(\mathbb {R})$ is relatively compact in the complete metric space $(BC(\mathbb {R}, \mathbb X), D_{b})$;

• (vi) (Bochner’s criterion) f is B-almost periodic if and only if $\tilde {f}(\mathbb {R})$ is relatively compact in the complete metric space $(BC(\mathbb {R}, \mathbb X), D_{b})$.

Proof

It is absolutely similar to the proof for Banach space valued functions, see e.g. [3, pp. 7–9].

Now, we are in position to prove the following sufficient condition for B-almost periodicity in p-Fréchet spaces, 0 < p < 1. □

Theorem 9.8

Let $f\in BC(\mathbb {R}, \mathbb X)$ . Let us suppose that there exists a relatively dense set of real numbers (s _n), such that

• (i) the set $\{f(s_{n}) ; n\in \mathbb {N} \}$ is relatively compact in the metric space (X, D) and

• (ii) for any $n, m\in \mathbb {N}$ , the relation

  $$\displaystyle \begin{aligned}||f(s_{n})-f(s_{m})||\ge c||f(\cdot + s_{n})-f(\cdot + s_{m})||{}_{b}\end{aligned}$$

  holds with c > 0 independent of n, m.

Then, f is almost periodic.

Proof

The inequality in statement together with Theorem 9.7, obviously implies

$$\displaystyle \begin{aligned}D[f(s_{n}), f(s_{m})]=||f(s_{n})-f(s_{m})||\ge c||\tilde{f}(s_{n})-\tilde{f}(s_{m})||{}_{b}=cD_{b}[\tilde{f}(s_{n}), \tilde{f}(s_{m})].\end{aligned}$$

Since by hypothesis, the set $\{f(s_{n}) ; n\in \mathbb {N} \}$ is relatively compact in the metric space (X, D), it has a convergent subsequence $(f(s^{\prime }_{n}))_{n}$, which therefore is a Cauchy sequence in the complete metric space (X, D), so it is convergent. The above inequality implies that $(\tilde {f}(s^{\prime }_{n}))_{n}$ is also a Cauchy sequence in the complete metric space $(C(\mathbb {R}, \mathbb X), D_{b})$, so it is convergent. Combined with Theorem 3.11(iv), it follows that $\tilde {f}$ is almost periodic, which combined with Theorem 3.11(iii), implies that f is almost periodic. The theorem is proved. □

2 Weakly Almost Periodic Functions

In what follows, we will consider the concept of weakly almost periodicity, at least in the cases of l ^p and H ^p spaces, 0 < p < 1. Indeed, according to Remark 9.13(1) in Sect. 2, the dual spaces (l ^p)^∗ and (H ^p)^∗ are non-null. In addition, since $\{e_{i}, i\in \mathbb {N}\}$, with e _i = (δ _i,n)_n ∈ l ^p, δ _i,n, Kronecker’s symbol, is a basis in l ^p (see e.g. [5], p. 20), and since any $e_{i}^{*}:l^{p}\to \mathbb {R}$ is linear and continuous (see e.g. [5], p. 12, Theorem 1.8), it easily follows that {p _φ;φ ∈ (l ^p)^∗}, with p _φ(x) = |φ(x)|, for all x ∈ l ^p, defines a sufficient family of seminorms on l ^p, which evidently induces a weak topology on l ^p, namely a locally convex Hausdorff topology on l ^p.

Also, since according to e.g. [37, p. 35], the point evaluations $\varphi _{z}(f)=f(z), z\in \mathbb {D}$, satisfy φ _z ∈ (H ^p)^∗, for all $z\in \mathbb {D}$, again it easily follows that {p _φ(x);φ ∈ (H ^p)^∗} with p _φ(x) = |φ(x)|, for all x ∈ H ^p, defines a sufficient family of seminorms on H ^p, which evidently induces a locally convex Hausdorff (weak) topology on H ^p.

Definition 9.9

Let X = l ^p or X = H ^p with 0 < p < 1. A function $f:\mathbb {R}\to X$ is called weakly almost periodic, if $f:\mathbb {R}\to X$ is continuous and almost periodic, considering X endowed with the (weak) locally convex topology as above (see e.g. [2, pp. 159–160], or [7, 8]).

Remark 9.10

Obviously that Definition 9.9 has no sense for the p-Fréchet space L ^p[0, 1], 0 < p < 1, whose dual is {0}.

Theorem 9.11

Let X = l ^p or X = H ^p, 0 < p < 1. The necessary and sufficient condition that the function $f:\mathbb {R}\to X$ be weakly almost periodic is that for any φ ∈ X ^∗ , the numerical function $h:\mathbb {R}\to \mathbb {R}$ , defined by h(t) = φ[f(t)], be almost periodic.

Proof

It is similar to the proof for Banach space-valued functions (see Theorem 6.1.7, p. 160 in [2]). □

Theorem 9.12

Let X = l ^p or X = H ^p, 0 < p < 1. The necessary and sufficient condition that the function $f:\mathbb {R}\to X$ be almost periodic is that f be weakly almost periodic and that $f(\mathbb {R})$ be relatively compact.

Proof

Since for any φ ∈ X ^∗ and all $t, \tau \in \mathbb {R}$, we have

$$\displaystyle \begin{aligned}|\varphi[f(t+\tau)]-\varphi[f(r)]|\le |||\varphi|||\cdot ||f(t+\tau)-f(t)||{}^{1/p},\end{aligned}$$

the usual (strong) almost periodicity and continuity of f immediately imply that f is weakly almost periodic and weakly continuous. This together with Theorem 3.6 immediately proves the necessity of theorem.

To prove the sufficiency, we would need the analogue for p-Fréchet space of the following result of Philips for Banach space (see the proof of Theorem 6.18, pp. 160–161 in [2]): from any bounded sequence (φ _n)_n in X ^∗ defined on a relatively compact subset A ⊂ X, one can extract a convergent subsequence on A. □

Remark 9.13

• (1) In the case when X is a p-Fréchet space endowed with the p-norm ||⋅||, 0 < p < 1, in [3, p. 102] (see also [1, p. 158]), the integral was introduced as follows. First, for a = a ₀ < a ₁ < … < a _n = b, a partition of [a, b], a step function on [a, b] is of the form $s(x)=\displaystyle \sum \limits _{k=0}^{n-1}y_{k}\cdot \chi _{[a_{k},a_{k+1})}(x)$ (where $\chi _{[a_{k},a_{k+1})}$ is the characteristic function of [a _k, a _k+1) and y _k ∈ X, k = 0, 1, …, n − 1), and its integral on [a, b] is defined by $\int _{a}^{b}s(x)dx=\displaystyle \sum \limits _{k=0}^{n-1}y_{k}(a_{k+1}-a_{k}).$ Then, since any continuous function f : [a, b] → X is uniformly continuous on [a, b], it is easy to show that it is the uniform limit on [a, b] of the sequences of step functions $s_{n}(x), n\in \mathbb {N}$ defined by

  $$\displaystyle \begin{aligned}s_{n}(x)=\displaystyle\sum_{k=0}^{n-1}f(a_{k})\cdot \chi_{[a_{k},a_{k+1})}(x), a_{k}=a+k\frac{b-a}{n}, k=0,1,\ldots,n-1,\end{aligned}$$

  and the integral of f will be defined by $\int _{a}^{b}f(x)dx\in X$, where

  $$\displaystyle \begin{aligned}\displaystyle\lim_{n\to \infty}\left\|\int_{a}^{b}f(x)dx - \int_{a}^{b}s_{n}(x)dx\right\|=0.\end{aligned}$$

  (It is easy to see that the above $\int _{a}^{b}f(x)dx$ does not depend on the sequence of step functions uniformly convergent to f.) Unfortunately, the fundamental theorem of calculus stated in [3, Theorem 2, pp. 104–105] (see also [10], pp. 161–162) seems to be not valid in general, since for a continuous function f : [a, b] → X, for the integral $F(t)=\int _{a}^{t}f(x)dx$, we have

  $$\displaystyle \begin{aligned}\left\|\frac{F(t+h)-F(t)}{h}-f(t)\right\|=\left\|\frac{\int_{t}^{t+h}[f(x)-f(t)]dx}{h}\right\|,\end{aligned}$$

  but we do not get, in general, the estimate

  $$\displaystyle \begin{aligned}\left\|\int_{t}^{t+h}[f(x)-f(t)]dx\right\|\le |h|{}^{p}||f(u)-f(t)||,\end{aligned}$$

  with u between t and t + h, as claimed in [10], p. 162 (which would imply that $\displaystyle \lim _{h\to 0}\left \|\frac {F(t+h)-F(t)}{h}-f(t)\right \|=0$). As a first consequence, it follows that the implication “f ^′ uniformly continuous and f B-almost periodic imply f ^′ B-almost periodic” does not hold, although in the case of Banach space-valued functions, it is valid (see e.g. [3, Theorem VI, p. 6]).

• (2) One could also adopt the more particular definition (of Riemann-type) for the integral on [a, b] of a function f : [a, b] → X, as unique limit of all the Riemann sums $\displaystyle \sum \limits _{k=0}^{n-1}f(\xi _{k})(a_{k+1}-a_{k})$, with ξ _k ∈ [a _k, a _k+1]. Unfortunately, for this kind of integral too, the property ||λx|| = |λ|^p||x||, where 0 < p < 1, produces a bad estimate for the difference between the Riemann sums attached to two functions f, g : [a, b] → X, namely

  $$\displaystyle \begin{aligned} \begin{array}{rlll} &&\left\|\displaystyle\sum_{k=0}^{n-1}f(\xi_{k})(a_{k+1}-a_{k})-\displaystyle\sum_{k=0}^{n-1}g(\xi_{k})(a_{k+1}-a_{k})\right\|\\ &&\quad \le\displaystyle\sum_{k=0}^{n-1}(a_{k+1}-a_{k})^{p}\left\|f(\xi_{k})-g(\xi_{k})\right\|\\ &&\quad \not\le \displaystyle\sum_{k=0}^{n-1}(a_{k+1}-a_{k})||f(\xi_{k})-g(\xi_{k})|| \end{array} \end{aligned}$$

  (in fact, since 0 < p < 1, for a _k+1 − a _k ≤ 1, we have (a _k+1 − a _k)^p ≥ (a _k+1 − a _k), which is the case for n sufficiently large). This fact that has the similar effect as for the first integral, namely the fundamental theorem of calculus for this second integral also does not hold.

• (3) From Remarks 9.13(1) and (2), it is evident that for a continuous f : [a, b] → X, the inequality

  $$\displaystyle \begin{aligned}\left\|\int_{a}^{b}f(t)dt\right\|\le \int_{a}^{b}||f(t)||dt\end{aligned}$$

  does not hold.

Now, if we introduce (as in the case of Banach space-valued functions) the mean value

$$\displaystyle \begin{aligned}M\left( f\right) =\displaystyle\lim_{T\rightarrow+\infty}\frac{1}{T}\cdot\int_{0} ^{T}f\left( t\right) dt\in X ,\end{aligned}$$

where the limit is considered in the metric space $\left ( X, D\right )$ (i.e. there exists $M\left ( f\right ) \in X$ with $\displaystyle \lim _{T\rightarrow +\infty }D\left ( M\left ( f\right ) ,\frac {1}{T}\cdot \int _{0}^{T}f\left ( t\right ) dt\right ) =0$), then because of Remark 9.13, it seems that M(f) does not exist for any f ∈ AP(X), since in the proof for the case of Banach space-valued functions, the inequality mentioned in Remark 9.13(3) is essential. This has as an effect the fact that, in general, one cannot attach Fourier series to a function f ∈ AP(X) and the fact that the almost periodicity of f does not imply the approximation property mentioned in Definition 3.1(iii).

• (4) In [6], a theory of the semigroups of linear and continuous operators is developed. As one of the applications, it was obtained that the initial value problem in the p-Fréchet space X, 0 < p < 1,

  $$\displaystyle \begin{aligned}\frac{du(t)}{dt}=A[u(t)], \;t\ge 0, u(0)=x\in X\end{aligned}$$

  (where A : X → X is linear and continuous) has as the unique solution u(t) = T(t)(x), with $T(t)(x)=\displaystyle \lim _{n\to \infty }\left (I-\frac {t}{n}A\right )^{-n}(x)$, the limit being in the p-norm in X. On the other hand, taking into account Remarks 9.13(1) and 9.13(2), it follows that the inhomogeneous Cauchy problem

  $$\displaystyle \begin{aligned}\frac{du(t)}{dt}=A[u(t)]+f(t), t\ge 0, u(0)=x\in X,\end{aligned}$$

  in general, seems to not have mild solution, in the sense that, even if we can define it as usual, it does not satisfy the differential equation.

• (5) It is easy to construct almost periodic functions $f:\mathbb {R}\to X$ for which there exists M(f) and the fundamental theorem of calculus holds. Indeed, any f of the form c ⋅ g, where c ∈ X and $g:\mathbb {R}\to \mathbb {R}$ is almost periodic, satisfies these two requirements.

Remark 9.14

The results in this section are contributions from Gal and N’Guérékata [30].

3 Applications

Firstly, we illustrate the idea of propagation of almost periodicity from the input data to the solutions of a simple differential equation in a p-Fréchet space (X, ||⋅||). In this sense, we present the following.

Theorem 9.15

Let $f:\mathbb {R}\rightarrow \mathbb {R}$ be a usual almost periodic function and c ∈ X. Then, the function $y:\mathbb {R} \rightarrow X$ given by

$$\displaystyle \begin{aligned}y(t)=c\cdot\int_{-\infty}^{t}e^{u-t}f(u)du,\mathit{}t\in\mathbb{R},\end{aligned}$$

is B-almost periodic and satisfies the differential equation

$$\displaystyle \begin{aligned}y^{\prime}(t) + y(t)=c\cdot f(t),\end{aligned}$$

for all $t\in \mathbb {R}$.

Here, y ^′(t) is defined as usual, that is, the limit in the metric D(x, y) = ||x − y||, given by

$$\displaystyle \begin{aligned}y^{\prime}(t)=\displaystyle\lim_{h\to 0}\frac{y(t+h)-y(t)}{h}.\end{aligned}$$

Proof

Let us denote $F(t)=\int _{-\infty }^{t}e^{u-t}f(u)du$, $t\in \mathbb {R}$. By the classical theory, F is a usual almost periodic function. Then, by

$$\displaystyle \begin{aligned}||c\cdot F(t) - c\cdot F(t+\tau)||=|F(t)-F(t+\xi)|{}^{p}\cdot ||c||,\end{aligned}$$

it is immediate that y(t) = c ⋅ F(t) is B-almost periodic in the sense of Definition 3.1(i).

Since

$$\displaystyle \begin{aligned}F^{\prime}(t)=f(t)-\int_{-\infty}^{t}e^{u-t}f(u), \forall t\in\mathbb{R},\end{aligned}$$

it easily follows that $y^{\prime }(t)=c\cdot \left [ f(t)-\int ^{t}_{-\infty }e^{u-t} f(u)du\right ]$ and that y(t) satisfies the differential equation, which proves the theorem. □

Bibliographical Notes

The materials in this chapter are due to Gal and N’Guérékata [30]. As one can see, there are some open problems that need further investigations.