UNIFORM LIMIT POWER-TYPE FUNCTION SPACES

In literature of Fourier transforms and Wavelet transforms, the basic space is L2(R). From the point of view of signal analysis, a signal f ∈ L2(R) can only be transient (or “wavelets”). During recent years in some application areas, it has become more common to motivate a theory via persistent rather than transient signals (e.g., [16, 28, 30, 42]). To work on persistent signals, people have to seek a space different from L2(R). One important example of such spaces is (R), the space of almost periodic functions. People have developed a profound theory and applications for (R) (e.g., see [4, 5, 7– 15, 18, 19, 24, 26, 30, 32, 37, 38]). As in [30], a function f is called limit power if the limit


Introduction
In literature of Fourier transforms and Wavelet transforms, the basic space is L 2 (R). From the point of view of signal analysis, a signal f ∈ L 2 (R) can only be transient (or "wavelets"). During recent years in some application areas, it has become more common to motivate a theory via persistent rather than transient signals (e.g., [16,28,30,42]). To work on persistent signals, people have to seek a space different from L 2 (R). One important example of such spaces is Ꮽᏼ(R), the space of almost periodic functions. People have developed a profound theory and applications for Ꮽᏼ(R) (e.g., see [4, 5, 7-15, 18, 19, 24, 26, 30, 32, 37, 38]).
As in [30], a function f is called limit power if the limit exists. Denote by H 2 the set of all such functions. It is well known that Ꮽᏼ(R) ⊂ H 2 and so is the Besicovitch space B 2 [5], the completion of Ꮽᏼ(R) in H 2 . In fact, many useful persistent signals are in H 2 , for example, the bounded power signals studied in Wiener's generalized harmonic analysis [36]. However, H 2 is not a linear set. An example in [29] shows that H 2 is not closed under addition. The lack of closedness under addition caused some difficulties in Robust control (e.g., see [27]).
As [29] pointed out that except for some subsets of H 2 which are already known to be vector spaces (e.g., L 2 (R), { f ∈ L ∞ (R) : lim |t|→∞ f (t) exists}, Ꮽᏼ(R)), it is not clear whether a "nice" (e.g., Hilbert) large vector space could be defined.
Let us recall when the spaces mentioned above were invented. The latest one is B 2 invented by Besicovitch in 1926 (see [5]); a year earlier is Ꮽᏼ(R + ) invented by Bohr [8][9][10]; L 2 (R) was invented even earlier. We remark that the function set studied by Wiener mentioned above is not closed under addition either. Some generalizations of Ꮽᏼ(R), for example, the functions studied in [1,2,17,21,[32][33][34]38], are vector spaces. They are larger than Ꮽᏼ(R) in Ꮿ(R). However, they are the same with Ꮽᏼ(R) in H 2 . We remark that though H 2 is not linear, there have been Banach spaces containing H 2 , for example, the space B 2 proposed in [11] (in [28] for the discrete setting). However, [11,28] use lim instead of lim in (1.1) to construct the spaces. In many cases, lim is needed too. The background of [29] and related problems being pointed out by some authors (e.g., [27,28,30] and references therein) show real needs for new, larger, nice spaces in H 2 .
The purpose of the paper is to propose such spaces. One will see that the new spaces are so natural that they come from what we call generalized trigonometric polynomials in the same way as Ꮽᏼ(R) and B 2 come from trigonometric polynomials. One will also see that they are so huge that to compare Ꮽᏼ(R) and B 2 with them is the same as to compare one point with R + .
The layout of the paper is as follows. In the next section, we show the existence of a larger orthonormal basis. In Section 3, we develop a theory of uniform limit power functions in a way parallel to that of Ꮽᏼ(R) (e.g., [12,13]). In Section 4, we discuss the limit power type functions.

Orthonormal basis
It is well known that {e iλt } is a complete orthonormal basis in B 2 [5]. In this section, we consider the set where λ ∈ R and 0 < α < ∞. When α > 1, in radar and sonar terminology, the function e iλt α represents a chirp signal because it is reasonably well defined but steadily rising frequency. By analyzing f (t) = sin(πt 2 ), [25,Chapter 2] points out the fact that a chirp has a well-defined instantaneous frequency and ordinary Fourier analysis hides the fact. By using Windowed Fourier transform, the signal is reasonably well localized both in time and in frequency. In particular, when α = 2, the function e it 2 , being an underlying kernel (e.g., in oscillatory integrals, optics, etc.), has important applications; we refer the reader to [3,6,20,22,23,31,35] for details.
When α < 1, the function e iλt α behaves conversely. As {e iλt }, the set {e iλt α } is also orthonormal. We show this in the next two theorems.
To estimate I 2 , we have where M 2 is a constant which is independent of T and a ∈ [a 0 ,∞). It follows from (2.6)-(2.8) that uniformly with respect to a ∈ [a 0 ,∞), and therefore with respect to a ∈ R + , the proof is complete.
Proof. Put μ = 0 in Theorem 2.1 to get the conclusion.
Proof. First, we consider the case α = β and w = λ + μ = 0, As in the proof of Theorem 2.1, one has T a e i(λt α +μt β ) dt = I 3 + I 4 , (2.14) To estimate I 4 we have the following: where M 4 and M 5 are constants which are independent of T. It follows that as T → ∞. The proof is complete.
It follows from Theorems 2.1 and 2.3 that That is, the set {e iλt α } constitutes an orthonormal basis.
Remark 2.4. Since the domain of the function e iλt α in general is R + , we consider R + only in the paper. For special numbers of α, for example, α s are positive integers, the domain will be R. In this case all the results will hold for R.

Uniform limit power functions
We call the functions n k=1 a k e iλkt α (3.1) α-trigonometric polynomial, where a k ∈ C and λ k ∈ R. As Ꮽᏼ(R), we have the following definition.
Denote by ᐁᏸᏼ α (R + ) the set of all such functions.
One sees that when α = 1, ᐁᏸᏼ α (R + ) = Ꮽᏼ(R + ). Also one sees that ᐁᏸᏼ α (R + ) is the completion of α-trigonometric polynomial in C(R + ). Since the set of α-trigonometric polynomials are closed under addition, multiplication, and conjugation, so is the completion ᐁᏸᏼ α (R + ). Thus we have shown the following statement: ᐁᏸᏼ α (R + ) is a C *subalgebra of C(R + ) containing the constant functions.
Next, we discuss the Fourier expansion of f ∈ ᐁᏸᏼ α (R + ). First of all, we show that the mean exists.
exists. In the case of α ≥ 1, exists uniformly with respect to a ∈ R + .
C. Zhang and W. Liu 7 Proof. We first show the theorem in the case that f is an α-trigonometric polynomial. Let c k e iλkt α . (3.5) Then by Theorems 2.1 and 2.3, If f is an arbitrary function in ᐁᏸᏼ α (R + ) then for > 0 there exists an α-trigonometric polynomial P such that (3.2) holds. Since lim T→∞ (1/T) T 0 P (t)dt exists, we can find a number T 0 such that when T 1 ,T 2 > T 0 , It follows from (3.2) and the last inequality above that when T 1 ,T 2 > T 0 , Similarly, one shows the existence of the second limit. The proof is complete.
We call the limit in Theorem 3.2 the mean of f and denote it by M( f ). For λ ∈ R and f ∈ ᐁᏸᏼ α (R + ) since the function f e −iλt α is in ᐁᏸᏼ α (R + ), the mean exists for the function. We write As the proof for Ꮽᏼ(R + ) (see [12,13,18,26,37,38]), for a function f ∈ ᐁᏸᏼ α (R + ) the frequency set is countable (or finite). Let Freq( f ) = {λ k } and A k = a(λ k ). Thus f has an associated Fourier series and Parseval's equality holds: The unique theorem for almost periodic function is well known. That is, distinct almost periodic functions have distinct Fourier series. We point out that this is also true for ᐁᏸᏼ α (R + ). To show this we need to set up some correspondence between ᐁᏸᏼ α (R + ) and Ꮽᏼ(R + ). For the α-trigonometric polynomial P in Definition 3.1, let s = t α . Then P becomes trigonometric polynomial of s. That is, a k e iλks . (3.13) For f ∈ ᐁᏸᏼ α (R + ) define the function So, f ∈ Ꮽᏼ(R + ). Conversely, let h ∈ Ꮽᏼ(R + ). By the approximation theorem of Ꮽᏼ(R + ) for > 0, there exists a trigonometric polynomial n k=1 a k e iλks such that Let s = t α (t ∈ R + ) and let h(t) = h(t α ). It follows that a k e iλkt α < t ∈ R + . (3.17) Therefore, we have h ∈ ᐁᏸᏼ α (R + ). Thus (3.14) is the correspondence between ᐁᏸᏼ α (R + ) and Ꮽᏼ(R + ). Note the translate property of almost periodic function, that is, for > 0 there exists l > 0 with the property that any interval I ⊂ R + of length l has a number τ ∈ I such that By the correspondence (3.14), we have in fact already shown the following theorem. (1) f ∈ ᐁᏸᏼ α (R + ); (2) for > 0 there exists l > 0 with the property that any interval I ⊂ R + of length l has a number τ ∈ I such that f (t + τ) 1/α − f t 1/α < . (3.19) Furthermore, if f ∈ ᐁᏸᏼ α (R + ) then so is | f (·)|. Now, we make use of the unique theorem for f to get the same conclusion for f . Proof. Let f be the function in (3.14). So f (s) ≥ 0 and f (s 0 ) > 0, where s o = t α 0 . Since f ∈ Ꮽᏼ(R + ), one has uniformly with respect to a ∈ R + . To show the theorem we need to discuss two cases.
By the lemma above we are able to show the following unique theorem. For f (t) ∈ ᐁᏸᏼ α (R + ) since f (s) ∈ Ꮽᏼ(R + ), the function f has an associated Fourier series a k e iμks , (3.25) where a k = M( f (s)e −iμk s ). It is well known (e.g., see [12,13,38]) that f can be approxi- uniformly on R + . The following remark tells us an important conclusion. where a k ∈ C and λ k ∈ R, 1 ≤ k ≤ n. If in Definition 3.1 P is a generalized trigonometric polynomial, then the function f is also called uniform limit power and ᐁᏸᏼ(R + ) is denoted the set of all such functions. It is not difficult to show that ᐁᏸᏼ(R + ) is a Banach space and (3.10)-(3.12) are valid. For the question if an f ∈ ᐁᏸᏼ(R + ) can be approximated by the Bochner-Fejer polynomials, as well as how to construct the polynomials, we refer the reader to [40,41] for details. Also where the closure is taken in C(R + ). One can see how huge ᐁᏸᏼ(R + ) is by comparing with Ꮽᏼ(R + ).
Remark 3.7. As chirps, some existing results enable us to analyze and reconstruct f ∈ ᐁᏸᏼ(R + ). For example, by [30, Theorem 2.1] a windowed Fourier transform of f exists, by Theorem 2.2 of the same paper the transform satisfies some Parseval's relation, and by Theorem 2.4 of that paper again a generalized frame exists.
One more remark is needed to end the section.

Remark 3.8.
(1) The conclusion in the paragraph before Remark 2.4 is mentioned in [39, Section 2] without proof. Here we not only prove it in details, but we also distinguish the limits between the case α ≥ 1 and the case α < 1 in Theorems 2.1 and 2.3, respectively. (2) Also, the results in Section 3 are presented in [39, Section 2] in an abstract-like form. To convince the reader the correctness of these results, we present and prove them in details here.

Limit power type functions
In this section, we will define and investigate three types of limit power function which are corresponding to the three types of well-known almost periodic functions (e.g., see [1,2,17,21,31,33,34,38]). Let . A function f is called asymptotic limit power if where g ∈ ᐁᏸᏼ α (R + ) and ϕ ∈ C 0 (R + ). Denote by Ꮽᏸᏼ α (R + ) all such functions.
Theorem 4.2. Let f ∈ C(R + ). Then the following statements are equivalent: (3) for any > 0 there exists a bounded closed interval C = [0,a] and l > 0 such that any interval I ⊂ R + of length l has a number τ ∈ I with the property If we only require the set in Theorem 4.2(2) to be weakly compact, then we get the following concept. To get the decomposition of a function f ∈ ᐃᏸᏼ α (R + ), we introduce the following set: (4.5) where the closure is taken under weak topology in C(R + ). The following result is a correspondence in ᐃᏸᏼ α (R + ) to that in ᐃᏭᏼ(R + ).
Finally we give the following concept.
for all λ ∈ R, Theorem 3.5 implies that ᏼᏸᏼ α (R + ) is a direct sum of ᐁᏸᏼ α (R + ) and ᏼᏭᏼ 0 (R + ). Since the ranges R f and R f are the same (so Rg and R g, Rϕ and R ϕ) and Proof. Let { f n } ⊂ ᏼᏸᏼ α (R + ) be Cauchy. Since R f n ⊃ Rg n , {g n } is Cauchy too. Note ᐁᏸᏼ α (R + ) is closed in C(R + ), there exists g ∈ ᐁᏸᏼ α (R + ) such that g n − g → 0 as n → ∞. Since { f n − g n } is also Cauchy and ᏼᏭᏼ 0 (R + ) is closed in C(R + ), there exists ϕ ∈ ᏼᏭᏼ 0 (R + ) such that ϕ n − ϕ → 0 as n → ∞. Let f = g + ϕ. Then f ∈ ᏼᏸᏼ α (R + ) and f n − f → 0 as n → ∞. The proof is complete. Since one has the following inclusion relationship:

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009