# Almost-periodic function

A function representable as a generalized Fourier series. There are several ways of defining classes of almost-periodic functions, based respectively on notions of closure, of an almost-period and of translation. Each of these classes can be obtained as a closure, with respect to some metric, of the set of all finite trigonometric sums.

Let $D _ {G} [ f (x) , \phi (x) ]$ be the distance of two functions $f (x)$ and $\phi (x)$ in a metric space $G$ of real- or complex-valued functions on $\mathbf R$. In the following, $G$ will be one of the spaces $U$, $S _ {l} ^ {p}$, $W ^ {p}$, or $B ^ {p}$. Here $U$ is the set of continuous bounded functions on the real line with the metric

$$D _ {U} [ f (x) , \phi (x) ] = \ \sup _ {- \infty < x < \infty } \ | f (x) - \phi (x) | ;$$

and $S _ {l} ^ {p} , W ^ {p}$ and $B ^ {p}$ for $p \geq 1$ are the sets of functions that are measurable and whose $p$- th powers are integrable on every finite interval of the real line, the metrics being

$$D _ {S _ {l} ^ {p} } [ f (x) , \phi (x) ] =$$

$$= \ \sup _ {- \infty < x < \infty } \left [ \frac{1}{l} \int\limits _ { x } ^ { x+l } | f (x) - \phi (x) | ^ {p} d x \right ] ^ {1/p} ,$$

$$D _ {W ^ {p} } [ f (x), \phi (x) ] = \ \lim\limits _ {l \rightarrow \infty } D _ {S _ {l} ^ {p} } [ f ( x ) , \phi (x) ] ,$$

$$D _ {B ^ {p} } [ f (x) , \phi (x) ] = \left [ \overline{\lim\limits}\; _ {\tau \rightarrow \infty } \frac{1}{2 \pi } \int\limits _ {- \tau } ^ \tau | f (x) - \phi (x) | ^ {p} d x \right ] ^ {1/p} .$$

Let $T$ be the set of trigonometric polynomials

$$\sum _ { k=1 } ^ { N } a _ {k} e ^ {i \lambda _ {k} x } ,$$

where the $\lambda _ {k}$ are arbitrary real numbers and the $a _ {k}$ are complex coefficients, and let the symbol $H _ {G} (T)$ denote the closure of $T$ in $G$. The classes $H _ {U} (T) = U$- a.p., $\overline{H}\; _ {S _ {l} ^ {p} } (T) = S _ {l} ^ {p}$- a.p., $H _ {W ^ {p} } (T) = W ^ {p}$- a.p. and $H _ {B ^ {p} } = B ^ {p}$- a.p. denote, respectively, the classes of uniformly almost-periodic functions, or Bohr almost-periodic functions, of Stepanov almost-periodic functions, of Weyl almost-periodic functions and of Besicovitch almost-periodic functions. These classes of almost-periodic functions are invariant under addition. Together with $f (x)$, each class also contained the functions $\overline{f}\; (x)$, $| f (x) |$ and $f (x) e ^ {i \lambda x }$, where $\lambda$ is a real number. The metrics $D _ {S _ {l} ^ {p} } [ f (x) , \phi (x) ]$ are topologically equivalent for all values of $l$; therefore it may be assumed that $l = 1$. Let $S _ {1} ^ {p}$- a.p. $= S ^ {p}$- a.p., $S ^ {1}$- a.p. $= S$- a.p., and $B ^ {1}$- a.p. $= B$- a.p.. Then

$$U - \textrm{ a }.p. \subset \ S ^ {p} - \textrm{ a }.p. \subset W ^ {p} - \textrm{ a }.p. \subset \ B ^ {p} - \textrm{ a }.p. ,\ \ p \geq 1 .$$

If $p _ {1} < p _ {2}$ and $p _ {1} \geq 1$, then

$$S ^ {p _ {2} } - \textrm{ a }.p. \subset \ S ^ {p _ {1} } - \textrm{ a }.p. ,\ \ W ^ {p _ {2} } - \textrm{ a }.p. \subset \ W ^ {p _ {1} } - \textrm{ a }.p. ,$$

$$B ^ {p _ {2} } - \textrm{ a }.p. \subset B ^ {p _ {1} } - \textrm{ a }.p. .$$

For every $f (x) \in B$- a.p., the mean value

$$M \{ f (x) \} = \ \lim\limits _ {\tau \rightarrow \infty } \ \frac{1} \tau \int\limits _ { 0 } ^ \tau f (x) d x$$

exists. The function $a ( \lambda , f ) = M \{ f (x) e ^ {- i \lambda x } \}$, where $\lambda$ is a real number, differs from zero only on a countable set of values of $\lambda$; any enumeration of this set is called the sequence $\{ \lambda _ {k} \}$, $k = 1 , 2 \dots$ of Fourier exponents of $f (x)$.

The numbers $A _ {\lambda _ {k} } = a ( \lambda _ {k} , f )$ are called the Fourier coefficients of $f (x)$. With a function $f (x)$ in any of the classes defined above one can associate its Fourier series

$$f (x) \sim \ \sum _ { k } A _ {\lambda _ {k} } e ^ {i \lambda _ {k} x } .$$

For $f (x) \in B ^ {2}$- a.p. one has the Parseval equality

$$M \{ | f (x) | ^ {2} \} = \ \sum _ { k } | A _ {\lambda _ {k} } | ^ {2} .$$

The Riesz–Fischer theorem can be generalized to the class $B ^ {p}$- a.p.: Let $\{ \lambda _ {k} \}$, $k = 1 , 2 \dots$ be arbitrary real numbers, and let $\{ A _ {k} \}$, $k = 1 , 2 \dots$ be complex numbers for which $\sum _ {k=1} ^ \infty | A _ {k} | < \infty$. Then there is an $f (x) \in B ^ {2}$- a.p. which has the trigonometric series $\sum _ {k} A _ {k} e ^ {i \lambda _ {k} x }$ as its Fourier series.

There is also a uniqueness theorem: If two functions $f (x) \in H _ {G} (T)$ and $\phi (x) \in H _ {G} (T)$ have the same Fourier series, then

$$D _ {G} [ f (x) ,\ \phi (x) ] = 0 .$$

In particular, for uniformly almost-periodic functions the uniqueness theorem states that $f (x) = \phi (x)$( for Stepanov almost-periodic periodic functions: almost-everywhere). A uniqueness theorem in the same sense as for Fourier–Lebesgue series of $2 \pi$- periodic functions does not hold for Weyl or Besicovitch almost-periodic functions.

The classes of uniformly almost-periodic and of Stepanov almost-periodic functions are, respectively, non-trivial extensions of the class of continuous $2 \pi$- periodic functions on $\mathbf R$ and the class of $2 \pi$- periodic integrable functions on the interval $[ 0 , 2 \pi ]$. For these classes of almost-periodic functions the uniqueness theorem remains valid.

A consequence of the definition of the classes of almost-periodic functions through the concept of closure is the approximation theorem: For every almost-periodic function $f (x)$ from $U$( or $S ^ {p}$ or $W ^ {p}$) and every $\epsilon > 0$ there is a finite trigonometric polynomial $P (x)$ in $T$, satisfying the inequality

$$D _ {U} [ f (x) , p (x) ] < \epsilon$$

$$(D _ {S ^ {p} } [ f (x) , P (x) ] < \epsilon ,\ D _ {W ^ {p} } [ f (x) , P (x) ] < \epsilon ) .$$

The approximation theorem may serve as a starting point of the definition of various classes of almost-periodic functions. The approximating polynomials $P (x)$ may contain "extraneous" exponents, i.e. exponents different from the Fourier exponents of $f (x)$. However, important for some applications of the approximation theorem is the fact that the exponents different from the Fourier exponents of $f (x)$ can be avoided in $P (x)$.

In connection with the representability of almost-periodic functions by generalized Fourier series, the problem of convergence criteria for these series arises and various summation methods for generalized Fourier series (the Bochner–Fejér method, etc.) become meaningful. Thus, the following criteria have been obtained: absolute convergence of a generalized Fourier series if the Fourier exponents are linearly independent; uniform convergence of a Fourier series when $| \lambda _ {k} | \rightarrow \infty$ as $k \rightarrow \infty$ or when $\lambda _ {k} \rightarrow 0$ as $k \rightarrow \infty$.

The importance of criteria for uniform convergence in the theory of almost-periodic functions is emphasized by the following theorem: If a trigonometric series $\sum _ {k} a _ {k} e ^ {i \lambda _ {k} x }$ is uniformly convergent on the entire real line, then it is the Fourier series of its sum $S (x) \in U$- a.p.. Corollary: There exists uniformly almost-periodic functions with an arbitrary countable set of Fourier exponents. If particular, the Fourier exponents of a uniformly almost-periodic function may have finite limit points or may even be everywhere dense.

Other definitions of almost-periodic functions of the above classes rely on the concept of an almost-period and generalizations thereof.

Besides the concept of closure or that of an almost-period, the concept of a translation can also be used for the definition of almost-periodic functions. Thus, a function $f (x)$ is uniformly almost-periodic if and only if every infinite sequence of functions $f ( x + h _ {1} ) , f ( x + h _ {2} ) \dots$ where the translation numbers $h _ {1} , h _ {2} \dots$ are arbitrary real numbers, contains a uniformly convergent subsequence. This definition serves as a starting point in considering almost-periodic functions on groups.

The main results in the theory of almost-periodic functions remain valid if one considers the concept of a generalized translation. Other generalizations are possible and useful: almost-periodic functions with values in an $n$- dimensional space or in a Banach or metric space, and analytic or harmonic almost-periodic functions.

#### References

 [1] H. Bohr, "Almost-periodic functions" , Chelsea, reprint (1947) (Translated from German) [2] A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932) [3] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) [4] N.P. Kuptsov, "Direct and converse theorems of approximation theory and semigroups of operators" Russian Math. Surveys , 32 : 4 (1968) pp. 115–177 Uspekhi Mat. Nauk , 23 : 4 (1968) pp. 117–178 [5] W. Rudin, "Fourier analysis on groups" , Benjamin (1962) [6] B.M. Levitan, "Generalized translation operators and some of their applications" , Israel Program Sci. Transl. (1964) (Translated from Russian) [7] A.M. Krasnosel'skii, "Non-linear almost-periodic oscillations" , Wiley (1973) (Translated from Russian)

The theory of almost-periodic functions was initiated by H. Bohr, who developed the notion of a uniformly almost-periodic function in his study of Dirichlet series. This original definition was the one using $\epsilon$- almost-periods and the approximation theorem was one of the main achievements of the theory. For a modern approach see [a4], Sect. 5 and [a10], Chapt. 1, 6. As to the equivalence of the approach starting from a certain structural property that is a generalization of pure periodicity, and on the other hand, the approach starting from approximation by trigonometric polynomials (also in the definition of the various classes of generalized almost-periodic functions), see [2], Chapt. 2. An interesting generalization not mentioned above are the Levitan almost-periodic functions (see [a5], where they are called $N$- almost periodic functions). A new approach to the theory of uniformly almost-periodic functions is given in [a1]; this has lead to the study of so-called almost-automorphic functions [a9], a class of functions closely related to the Levitan almost-periodic functions mentioned above [a7].