1.2 Almost-Periodic Functions

In this section, definitions and main results on almost-periodic (AP) functions and their generalizations are presented for both continuous-and discrete-time cases. For extensive treatments on almost-periodic functions, see (Besicovitch 1932), (Bohr 1933), and (Corduneanu 1989) for continuous-time, and (Corduneanu 1989, Chapter VII), (Jessen and Tornehave 1945), and (von Neumann 1934) for discrete-time.

1.2.1 Uniformly Almost-Periodic Functions

Definition 1.2.1 (Besicovitch 1932, Chapter 1). A function z(t), , is said to be uniformly almost-periodic if such that for any interval such that

(1.51)

The quantity is said translation number of z(t) corresponding to .

A set is said to be relatively dense in if such that the result is that D∩ I ≠ .

Thus, defined the set of the translation numbers of z(t) corresponding to

(1.52)

according to Definition 1.2.1, the function z(t) is uniformly almost periodic if and only if the set is relatively dense in . That is, there are many translation numbers of z(t) corresponding to .

Theorem 1.2.2 (Besicovitch 1932, Chapter 1). Any uniformly AP function is the limit of a uniformly convergent sequence of trigonometric polynomials in t (generalized Fourier series):

(1.53)

where the frequencies α A, with A countable set of possibly incommensurate reals and possibly containing cluster points, and

(1.54)

with the limit independent of t₀. Thus, z(t) is bounded and uniformly continuous.

Theorem 1.2.3 (Besicovitch 1932, Chapter 1). For any uniformly AP function the following Parseval's equality holds

(1.55)

A function z(t) is periodic with period T₀ > 0 if T₀ is the smallest nonzero value such that

(1.56)

Thus, periodic functions are obtained as special case of almost-periodic functions with independent of , . In such a case, the frequencies of the set A are all multiple of a fundamental frequency 1/T₀, that is, and (1.53) is the ordinary Fourier series expansion of the periodic function z(t).

An example of uniformly AP function which is not periodic is

(1.57)

Both cosines are periodic functions with periods T₀ and , respectively. However, their sum is not periodic since the ratio of the two periods T₀ and is the irrational number .

The functions defined in Definition 1.2.1 and characterized in Theorem 1.2.1 are called almost-periodic in the sense of Bohr (Bohr 1933, paragraphs 84–92) or, equivalently, uniformly almost periodic in the sense of Besicovitch (Besicovitch 1932, Chapter 1), or, equivalently, almost-periodic with respect to the sup norm. More general classes of almost-periodic functions, including possibly discontinuous functions, are treated in (Besicovitch 1932, Chapter 2) and the following sections.

1.2.2 AP Functions in the Sense of Stepanov, Weyl, and Besicovitch

The almost-periodicity property can be defined with respect to the following norms or seminorms, with p ≥ 1, (Besicovitch 1932, Chapter 2):

Functions belonging to with finite Besicovitch B^p-seminorm form a seminormed space called Marcinkiewicz space .

Let denote any of the above defined (semi)norms, that is, , , or . For each (semi)norm, a definition of almost-periodicity can be given.

Definition 1.2.4 , W^p, and B^p Almost Periodicity (Besicovitch (1932) Chapter 2). A function z(t) is said to be G^p almost-periodic (G^p-AP), p ≥ 1, if ∀ > 0 ∃ > 0 such that for any interval such that

(1.61)

Specifically, if , then z(t) is said -AP; If , then z(t) is said ; If , then z(t) is said -AP.

Theorem 1.2.5 (Besicovitch 1932, Chapter 2). Any G^p-AP function is G^p-bounded () and is the G^p-limit of a sequence of trigonometric polynomials in t:

(1.62)

where A_n is an increasing sequence of countable sets such that and . The coefficients z_α of the generalized Fourier series are given by (1.54).

In (Besicovitch 1932, p. 74) it is shown that: If , then z₁(t) = z₂(t) a.e.; If , then z₁(t) and z₂(t) may differ at a set of points of finite and even of infinite measure. In addition, for p ≥ 1 the result is that (Besicovitch 1932, p. 73) .

Theorem 1.2.6 (Besicovitch 1932, Chapter 2). For any G²-AP function the following Parseval's equality holds

(1.63)

Further generalizations of almost-periodic functions can be found in (Besicovitch 1932, Chapter 2), (Bohr 1933, paragraphs 94–102), and (Corduneanu 1989, Chapter VI).

1.2.3 Weakly AP Functions in the Sense of Eberlein

Definition 1.2.7 Weakly Almost-Periodic Functions (Eberlein 1949, 1956). A continuous and bounded function z(t) is said to be weakly almost-periodic (w.a.p.) (in the sense of Eberlein) if the set of translates z(t + τ), , is (conditionally) weakly compact in the set of continuous and bounded functions .

Examples of w.a.p. functions are the uniformly almost periodic functions (in the sense of Definition 1.2.1), the positive definite functions (hence Fourier-Stieltjes transforms), and functions vanishing at infinity (Eberlein 1949, Theorems 11.1 and 11.2). A w.a.p. function is uniformly continuous (Eberlein 1949, Theorem 13.1).

Theorem 1.2.8 (Eberlein 1956). Every w.a.p. function z(t) admits a unique decomposition

(1.64)

where z_uap(t) is a uniformly almost-periodic function in the sense of Definition 1.2.1 and z₀(t) is a zero-power function

(1.65)

Moreover, it results that

(1.66a)

(1.66b)

and, accordingly with (1.65) and using the notation of Theorem 1.2.5, the result is that

(1.67)

Theorem 1.2.9 (Eberlein 1956). For any w.a.p. function the following Parseval's equality holds

(1.68)

It is worthwhile emphasizing that the set of w.a.p. functions, unlike other classes of generalized AP functions, is closed under multiplication (Eberlein 1949). That is, the product of two w.a.p. functions is in turn a w.a.p. function.

Other definitions of w.a.p. functions different from Definition 1.2.7 are given in (Amerio and Prouse 1971, Chapter 3), (Corduneanu 1989, Section VI.5), (Zhang and Liu 2010).

Theorem 1.2.10 (Eberlein 1949, Theorem 15.1). Let z₁(t) and z₂(t) be w.a.p. functions. Then

(1.69)

exists and is a uniformly almost periodic function of t. A similar result also holds with different definitions of w.a.p. functions (Zhang and Liu 2010).

1.2.4 Pseudo AP Functions

Definition 1.2.11 (Ait Dads and Arino 1996). The function z(t) is said to be pseudo almost-periodic in the sense of Ait Dads and Arino, shortly , if it admits the (unique) decomposition

(1.70)

where z_uap(t) is a uniformly almost-periodic function in the sense of Definition 1.2.1 and z₀(t), referred to as the ergodic perturbation, is a Lebesgue measurable function such that

(1.71)

shortly .

The classes and are slight generalizations of the classes and , respectively, of the pseudo almost-periodic functions in the sense of Zhang (Zhang 1994, 1995), where z(t) and z₀(t) are assumed to be continuous and bounded.

Theorem 1.2.12 (Ait Dads and Arino 1996). Let be . It results that

(1.72a)

(1.72b)

Proposition 1.2.13 (Ait Dads and Arino 1996). If exists, then 0 and z(t) belong to the class of the asymptotically almost-periodic functions in the sense of Frechet.

Properties of asymptotically almost-periodic functions are given in (Lekow and Napolitano 2006, Section 6.2).

1.2.5 AP Functions in the Sense of Hartman and Ryll-Nardzewski

Definition 1.2.14 (Kahane 1962), (Andreas et al. (2006) Definition 7.1). The function is said to be almost periodic in the sense of Hartman, shortly , if,

(1.73)

exists and is finite.

Definition 1.2.15 (Kahane 1962), (Andreas et al. (2006) Definition 7.2). The function is said to be almost periodic in the sense of Ryll-Nardzewski, shortly , if,

(1.74)

exists uniformly with respect to and is finite.

Obviously, if , then and , but the converse is not true. That is, .

If or , then and the Fourier coefficients of the (generalized) Fourier series in (1.62) are coincident with those in (1.74). If , then (but not necessarily ) and the Fourier coefficients of the (generalized) Fourier series in (1.62) are (obviously) those in (1.73).

Theorem 1.2.16 (Kahane 1961), (Andreas et al. (2006) Theorem 7.5). The spectrum of , that is the set , is at most countable. Consequently, also the spectrum of is at most countable.

Theorem 1.2.17 (Urbanik 1962), (Kahane 1961). Let , p > 1. Then, the following unique decomposition holds

(1.75)

where z_Bap(t) is a B^p-AP function and with empty spectrum, that is

(1.76)

In particular, since uniformly AP functions in the sense of Definition 1.2.1 are special cases of B^p-AP functions, the function z_Bap(t) in decomposition (1.75) can reduce to a uniformly AP function.

Let us define the sets

(1.77)

(1.78)

Obviously R₀ ⊂ H₀.

Theorem 1.2.18 (Kahane 1962). Let be . Then, there exists such that . Let be . Then, there exists such that .

Theorem 1.2.19 (Kahane 1962). Let be . Then, there exist such that .

Theorem 1.2.20 (Kahane 1962). Let x(t) be uniformly continuous and bounded. Then, there exist uniformly continuous and bounded such that . In particular, x(t) can be uniformly almost periodic.

1.2.6 AP Functions Defined on Groups and with Values in Banach and Hilbert Spaces

Almost-periodic functions and their generalizations on groups are treated in (Corduneanu 1989, Chapter VII), (von Neumann 1934), (Casinovi 2009).

Almost-periodic functions with values in Hilbert spaces are treated in (Phong 2007). Further classes of AP functions with values in Banach spaces and a survey of their properties are presented in (Andreas et al. 2006), (Chérif 2011a,b).

1.2.7 AP Functions in Probability

Let , denoted shortly by z(t), be a random process defined on a probability space .

Definition 1.2.21 Random Functions Almost-Periodic in Probability (Corduneanu (1989) Sect. II.3). A random process z(t), , is called almost-periodic in probability if , , there exists such that for every set of length , say , there exists at least one number such that

(1.79)

The real number is said -almost period in probability.

Theorem 1.2.22 (Corduneanu 1989, Sect. II.3). Any random process AP in probability is bounded in probability and is the limit in probability of a sequence of random trigonometric polynomials in t. That is, ∀η > 0,

(1.80)

where A_n is an increasing sequence of countable sets of real numbers α and z_α(ω) are random variables.

1.2.8 AP Sequences

Definition 1.2.23 Discrete-Time Almost-Periodic Functions (Corduneanu 1989, Chapter VII), (Jessen and Tornehave 1945), (von Neumann 1934). A sequence z(n), , is said to be almost-periodic if such that for any set such that

(1.81)

The integer m is said to be the translation number of z(n) corresponding to .

Theorem 1.2.24 (Corduneanu 1989, Chapter VII), (von Neumann 1934). Every AP sequence is the limit of a sequence of trigonometric polynomials in n:

(1.82)

where the frequencies , with countable set with possibly incommensurate elements in [− 1/2, 1/2) and possibly containing cluster points,

(1.83)

with the limit independent of n₀ and

(1.84)

Thus, z(n) is bounded.

A sequence z(n) is periodic with period N₀ if N₀ is the smallest non-zero integer such that

(1.85)

Thus, periodic sequences are obtained as a special case of almost-periodic sequences with m = kN₀ independent of , . In such a case, is the finite set {0, 1/N₀, ..., (N₀ − 1)/N₀} or any equivalent set k₀ + A {k₀, k₀ + 1/N₀, ..., k₀ + (N₀ − 1)/N₀} with k₀ integer, and (1.82) is the discrete Fourier series (DFS) of z(n).

In continuous-time, the complex sinewave is periodic with period and the polynomial phase signal , γ > 1, is not almost periodic. Complex discrete-time sinewaves and polynomial phase sequences require more attention. The sequence with , p, q relative prime integers or co-prime (that is, they have no common positive divisor other than 1 or, equivalently, their greatest common divisor is 1 or, equivalently, p/q is an irreducible fraction), is periodic with period q. In contrast, the sequence with , is almost periodic (not periodic). For every positive integer L ≥ 2, the sequence with , p, q relative prime integers, is periodic with period q whereas it is not almost periodic for .

1.2.9 AP Sequences in Probability

Definition 1.2.25 Almost-Periodic Random Sequences in Probability (Han and Hong 2007). A random sequence z(n), , is called almost-periodic in probability if , , there exists such that for every set of length , say , there exists at least one number such that

(1.86)

The integer m_,η is said to be (, η)-almost period in probability.