Periodic wavelet frames and time-frequency localization
aa r X i v : . [ m a t h . C A ] M a r Periodic wavelet frames and time-frequency localization ∗ Elena A. Lebedeva, J¨urgen [email protected], [email protected]
Abstract
A family of Parseval periodic wavelet frames is constructed. The family has optimal time-frequencylocalization (in the sense of the Breitenberger uncertainty constant) with respect to a family parameter andit has the best currently known localization with respect to a multiresolution analysis parameter.
Keywords: periodic wavelet, scaling function, Parseval frame, tight frame, uncertainty principle, Poissonsummation formula, localization
MSC[2010]
In recent years the wavelet theory of periodic functions has been continuously refined. First, periodic waveletswere generated by periodization of wavelet functions on the real line (see, for example, [6]). A wider and morenatural approach providing a flexibility on a theoretical front and in applications is to study periodic waveletsdirectly using a periodic analog of a multiresolution analysis (MRA). The concept of periodic MRA is introducedand discussed in [14, 18, 19, 20, 21, 26, 27, 29]. In [9], a unitary extension principle (UEP) for constructingParseval wavelet frames is rewritten for periodic functions (see Theorem 1). The approach is developed furtherin [8].In this paper we focus on a property of good localization of both periodic wavelet functions and their Fouriercoefficients. The quantitative characteristic of this property is an uncertainty constant (UC). Originally, theconcept of the UC was introduced for the real line case in 1927 (see Definition 1) by Heisenberg in [12].Its periodic counterpart was introduced in 1985 by Breitenberger in [3] (see Definition 2). The smaller UCcorresponds to the better localization. In both cases there exists a universal lower bound for the UC (theuncertainty principle). In non-periodic setup the minimum is attained on the Gaussian function. But thereis no periodic function attending the lower bound. So, to find a sequence of periodic functions having anasymptotically minimal UC and some additional setup, for example a wavelet structure, is a natural concern.There is a connection between the Heisenberg and the Breitenberger UCs for wavelets. In [23] it is provedthat for periodic wavelets generated by periodization (see the definition in Section 4) of a wavelet function on thereal line the periodic UC tends to the real line UC of the original function as a parameter of periodization tendsto infinity. It would be a possible way to construct an optimal periodic wavelet system using the periodizationof a wavelet system on the real line. However, in [2] and [1] the following result is proven: If a real line function ψ generates a wavelet Bessel set and the frequency center ω , c ψ = ( ψ ′ , ψ ) L ( R ) = 0 (see notation ω , c ψ inDefinition 1), then the Heisenberg UC is greater or equal to 3 / . Moreover, it is unknown if there exists areal line orthonormal wavelet basis or tight frame possessing the Heisenberg UC less than 2 . . This valueis attained for a Daubechies wavelet [7]. The smallest possible value of the Heisenberg UC for the family ofthe Meyer wavelets equals to 6 .
874 [17]. It is well known [5] that the Heisenberg UC of the Battle-Lemarieand the Daubechies wavelets tends to infinity as their orders grow. A set of real line orthogonal wavelet baseswith the uniformly bounded Heisenberg UCs as their orders (smoothness) grow is constructed in [15, 16]. Onthe other hand, there are examples of real line wavelet frames possessing asymptotically optimal UC such asnonorthogonal B-spline wavelets [28] and their generalizations [11]. However, these frames are not tight and weare looking for an orthogonal basis or tight frame. We will discuss a particular issue of periodization in Section4. Some papers dealing with periodic UCs directly include [10, 22, 24, 25]. For the first time in [25] periodicUCs uniformly bounded with respect to an MRA parameter are computed for so-called trigonometric wavelets ∗ The first author is supported by grant of President RF MK-1847.2012.1, by RFBR 12-01-00216-a, and by DAAD scholarshipA/08/79920. { ϕ h } h> is constructed, namely U C ( ϕ h ) < / √ h/ . Later, ϕ h is used as a scaling function to generatea stationary interpolatory MRA ( V n ). For the corresponding wavelet functions ψ n,h the UC is optimal for afixed space V n , but the estimate is nonuniform with respect to n, namely U C ( ψ n,h ) < / . n √ h. Nothingchanges after orthogonalization:
U C ( ψ ⊥ n,h ) < / . n √ h, U C ( ϕ ⊥ n,h ) < / n √ h. The main contribution of this paper is Theorem 4, where we construct a family of scaling sequencesΦ = (cid:8) ( ϕ aj ) j : a > (cid:9) generating a family of wavelet sequences Ψ = (cid:8) ( ψ aj ) j : a > (cid:9) corresponding to anonstationary periodic MRA as it is defined in [8], [14], and [27]. For a fixed level j of the MRA ( V j ), similarto the construction in [22], the UCs of ϕ aj and ψ aj are asymptotically optimal, that islim a →∞ sup j ∈ N U C ( ϕ aj ) = 12 , lim a →∞ U C ( ψ aj ) = 12 . But now, for a fixed value of the parameter a > , the scaling sequence has the asymptotically optimal UC, andthe wavelet sequence has the smallest currently known value of the UC for the periodic wavelet frames setup,that is lim j →∞ sup a> U C ( ϕ aj ) = 12 , lim j →∞ U C ( ψ aj ) = 32 . As it is indicated above, the functions constructed in [22] do not have this property.This issue partly answers the question stated in [22] whether there exists a translation-invariant basis of awavelet space W j which is asymptotically optimal independent of the MRA level j. In Theorem 4 we get anaffirmative answer for the case of scaling functions corresponding to tight wavelet frames. The case of waveletbasis is an open problem and it is a task for future work. In this direction, some useful properties of shiftedGaussian are discussed in [13]. We will consider the particular issue of wavelet sequences in Section 4.
Let L (0 ,
1) be the space of all 1-periodic square-integrable complex-valued functions, with inner product( · , · ) given by ( f, g ) := R f ( x ) g ( x ) d x for any f, g, ∈ L (0 , , and norm k · k := p ( · , · ) . The Fourier seriesof a function f ∈ L (0 ,
1) is defined by P k ∈ Z b f ( k )e π i kx , where its Fourier coefficient is defined by b f ( k ) = R f ( x )e − π i kx d x. Let H be a separable Hilbert space. If there exist constants A, B > f ∈ H the followinginequality holds A k f k ≤ P ∞ n =1 | ( f, f n ) | ≤ B k f k , then the sequence ( f n ) n ∈ N is called a frame for H. If A = B (= 1) , then the sequence ( f n ) n ∈ N is called a tight frame (a Parseval frame) for H. In addition, if k f n k = 1 for all n ∈ N , then the system forms an orthonormal basis. More information about frames can befound in [4].In the sequel, we use the following notation f j,k ( x ) := f j ( x − − j k ) for a function f j ∈ L (0 , . Considerfunctions ϕ , ψ j ∈ L (0 , , j = 0 , , . . . If the collection Ψ := (cid:8) ϕ , ψ j,k : j = 0 , , . . . , k = 0 , . . . , j − (cid:9) , forms a frame (or basis) for L (0 ,
1) then Ψ is said to be a periodic wavelet frame (or wavelet basis) for L (0 , . Let us recall the UEP for a periodic setting. We consider a case of one wavelet generator.
Theorem 1 ([9])
Let ϕ j ∈ L (0 , , j = 0 , , . . . , be a sequence of -periodic functions such that lim j →∞ j/ b ϕ j ( k ) = 1 . (1) Let µ jk be a two-parameter sequence such that µ jk +2 j = µ jk , and b ϕ j − ( k ) = µ jk b ϕ j ( k ) . (2) Let ψ j , j = 0 , , . . . , be a sequence of -periodic functions defined using Fourier coefficients b ψ j ( k ) = λ j +1 k b ϕ j +1 ( k ) , (3) where λ jk +2 j = λ jk and µ jk µ jk +2 j − λ jk λ jk +2 j − ! µ jk λ jk µ jk +2 j − λ jk +2 j − ! = (cid:18) (cid:19) . (4) Then the family
Ψ := (cid:8) ϕ , ψ j,k : j = 0 , , . . . , k = 0 , . . . , j − (cid:9) forms a Parseval wavelet frame for L (0 , . ϕ j ) j , ( ψ j ) j , ( µ jk ) k , and ( λ jk ) k are called scaling sequence, wavelet sequence, mask andwavelet mask respectively. This setup generates a periodic MRA: By definition, put V j = span (cid:8) ϕ j,k ; k = 0 , . . . , j − (cid:9) for j ≥ . Then the sequence ( V j ) j ≥ is a periodic MRA.Let us recall the definitions of the UCs and the uncertainty principles. Definition 1 ([12])
The (Heisenberg) UC of f ∈ L ( R ) is the functional U C H ( f ) := ∆ f ∆ b f such that ∆ f := k f k − L ( R ) R R ( t − t f ) | f ( t ) | d t, ∆ b f := k b f k − L ( R ) R R ( ω − ω b f ) | b f ( ω ) | d ω,t f := k f k − L ( R ) R R t | f ( t ) | d t, ω b f := k b f k − L ( R ) R R ω | b f ( ω ) | d ω. Theorem 2 ([12])
Let f ∈ L ( R ) , then U C H ( f ) ≥ / , and the equality is attained iff f is the Gaussian. Definition 2 ([3])
Let f ( x ) = P k ∈ Z c k e π i kx ∈ L (0 , . The first trigonometric moment is defined as τ ( f ) := − π Z e π i x | f ( x ) | d x = − π X k ∈ Z c k ¯ c k +1 . The angular variance of the function f is defined by var A ( f ) := 14 π (cid:0)P k ∈ Z | c k | (cid:1) (cid:12)(cid:12)P k ∈ Z c k ¯ c k +1 (cid:12)(cid:12) − ! = k f k | τ ( f ) | − π . The frequency variance of the function f is defined by var F ( f ) := 4 π P k ∈ Z k | c k | P k ∈ Z | c k | − π (cid:0)P k ∈ Z k | c k | (cid:1) (cid:0)P k ∈ Z | c k | (cid:1) = k f ′ k k f k + ( f ′ , f ) k f k . The quantity
U C ( { c k } ) := U C ( f ) := p var A ( f )var F ( f ) is called the periodic (Breitenberger) UC . Theorem 3 ([3, 22])
Let f ∈ L (0 , , f ( x ) = C e π i kx , C ∈ R , k ∈ Z . Then U C ( f ) > / and there is nofunction such that U C ( f ) = 1 / . Since periodic wavelet bases and frames are nonstationary in nature and the UC has no extremal function,it is natural to give the following
Definition 3
Suppose that ϕ j ( ψ j ) is a scaling (a wavelet) sequence. Then the quantity lim sup j →∞ U C ( ϕ j ) (lim sup j →∞ U C ( ψ j )) is called the UC of the scaling (the wavelet) sequence . We say that a sequence of periodic functions ( f j ) j ∈ N has an optimal UC if lim j →∞ U C ( f j ) = 1 / . To justify the definition we note that since inf
U C ( f ) = 1 / , it follows that if lim sup j →∞ U C ( f j ) = 1 / , thenlim j →∞ U C ( f j ) = 1 / . So in the optimal case one can use lim j →∞ instead of lim sup j →∞ . In the following theorem we construct a family of periodic Parseval wavelet frames with the optimal UCs forscaling functions and currently the best known UCs for wavelets.
Theorem 4
There exists a family of periodic wavelet sequences Ψ a := { ( ψ aj ) j } a such that for any fixed a > the system { ϕ a } ∪ { ψ aj,k : j = 0 , , . . . , k = 0 , . . . , j − } forms a Parseval frame in L (0 , and lim j →∞ sup a> U C ( ϕ aj ) = 12 , lim a →∞ sup j ∈ N U C ( ϕ aj ) = 12 , (5)lim j →∞ U C ( ψ aj ) = 32 , lim a →∞ U C ( ψ aj ) = 12 . (6)3 ut ϕ a = 1 . Let ν j,ak be a sequence given by ν ,a = ν ,a = p / and ν j,ak := exp (cid:16) − k + a j ( j − a (cid:17) , k = − j − + 1 , . . . , j − , r − exp (cid:16) − k − j − ) + a ) j ( j − a (cid:17) , k = 2 j − + 1 , . . . , × j − , (7) and extended j -periodic with respect to k . Furthermore, we define b ξ aj ( k ) := Q ∞ r = j +1 ν r,ak . Then the scalingsequence, masks, wavelet masks and wavelet sequence are defined respectively as c ϕ aj ( k ) := 2 − j/ b ξ aj ( k ) , µ j,ak := √ ν j,ak ,λ j,ak := e πi − j k µ j,ak +2 j − , b ψ aj ( k ) := λ j +1 ,ak b ϕ aj +1 ( k ) . (8) Remark 1
The UC is a homogeneous functional, that is
U C ( αf ) = U C ( f ) for α ∈ R , so U C ( ϕ aj ) = U C (2 − j/ ξ aj ) = U C ( ξ aj ) and in the sequel we prove the equalities lim j →∞ sup a> U C ( ξ aj ) = 1 / and lim a →∞ sup j ∈ N U C ( ξ aj ) = 1 / instead of (5). By analogy, let η aj := 2 j/ ψ aj , then U C ( ψ aj ) = U C ( η aj ) . To prove Theorem 4, we need some technical Lemmas.
Lemma 1
The UC is a continuous functional with respect to the norm k f k W := k f k + k f ′ k . Proof.
Indeed, τ ( f ) and ( f ′ , f ) are continuous with respect to this norm. Using the Cauchy-Bunyakovskiy-Schwarz inequality, we immediately get12 π | τ ( f ) − τ ( g ) | ≤ Z (cid:12)(cid:12) | f | − | g | (cid:12)(cid:12) = (cid:16)(cid:12)(cid:12)(cid:12) | f | − | g | (cid:12)(cid:12)(cid:12) , | f | + | g | (cid:17) ≤ (cid:13)(cid:13)(cid:13) | f | − | g | (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) | f | + | g | (cid:13)(cid:13)(cid:13) ≤ (cid:16) k f k + k g k (cid:17) k f − g k W ; | ( f ′ , f ) − ( g ′ , g ) | ≤ | ( f ′ , f − g ) | + | ( f ′ − g ′ , g ) | ≤ k f ′ k k f − g k + k f ′ − g ′ k k g k ≤ max n k f ′ k , k g k o k f − g k W . It remains to note that the UC continuously depends on k f k , k f ′ k , τ ( f ) , and ( f ′ , f ) . (cid:3) Lemma 2
Suppose α, β, γ ∈ R , m = 0 , , . . . , and < b < M, where M is an absolute constant, then X k ∈ Z ( αk + βk + γ ) m e − b ( αk + βk + γ ) = ( − m (cid:18) exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) r πbα (cid:19) ( m ) b m + exp (cid:18) − π − εbα (cid:19) O (1) , (9) as b → , where ε > is an arbitrary small parameter. Proof.
It is possible to change the order of summation and differentiation, so X k ∈ Z ( αk + βk + γ ) m e − b ( αk + βk + γ ) = ( − m X k ∈ Z e − b ( αk + βk + γ ) ! ( m ) b m = ( − m exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) X k ∈ Z exp − bα (cid:18) k + β α (cid:19) !! ( m ) b m . Using the Poisson summation formula for the function f ( t ) = e − bαt X k ∈ Z e − bα ( k − t ) = r πbα X k ∈ Z cos 2 πkt exp (cid:18) − π k bα (cid:19) , (10)with t = − β/ (2 α ) , and then differentiating m times with respect to b , we get( − m exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) r πbα X k ∈ Z cos (cid:18) πk β α (cid:19) exp (cid:18) − π k bα (cid:19)! ( m ) b m = ( − m (cid:18) exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) r πbα (cid:19) ( m ) b m + ( − m (cid:18) exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) r πbα (cid:19) ( m ) b m ∞ X k =1 cos (cid:18) πk β α (cid:19) exp (cid:18) − π k bα (cid:19) +( − m m X r =1 (cid:18) mr (cid:19) (cid:18) exp (cid:18) − b (cid:18) γ − β α (cid:19)(cid:19) r πbα (cid:19) ( m − r ) b m − r ∞ X k =1 cos (cid:18) πk β α (cid:19) exp (cid:18) − π k bα (cid:19)! ( r ) b r . r = 0 , . . . , m, we estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 cos (cid:18) πk β α (cid:19) e − π k / ( bα ) ! ( r ) b r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X k =1 Q r (cid:18) k, b (cid:19) e − π k / ( bα ) ≤ e − π / ( bα ) ∞ X k =1 Q r (cid:18) k, b (cid:19) exp (cid:18) − π ( k − M α (cid:19) = e − ( π − ε ) / ( bα ) O (1) , where Q r ( k, /b ) is a polynomial of degree 2 r in k, and 1 /b. We estimate summands of the form e − π / ( bα ) /b ξ , < ξ < m by e − π / ( bα ) /b ξ < exp (cid:16) − π − εbα (cid:17) . (cid:3) Lemma 3
Suppose η a, j ( t ) := P k ∈ Z b η a, j ( k )e π i kt , where b η a, j ( k ) := e π i2 − j − k s − exp (cid:18) − k + a )( j ( j + 1) a ) (cid:19) exp (cid:18) − k + a ( j + 1) a (cid:19) ; (11) then lim j →∞ U C ( η a, j ) = 3 / for any fixed a > and lim a →∞ U C ( η a, j ) = 1 / for any fixed j ∈ N . Proof.
We estimate the quantities (( η a, j ) ′ , η a, j ) , k η a, j k , k ( η a, j ) ′ k , and | τ ( η a, j ) | and then substitute theexpressions in Definition 2. Since | b η a, j ( k ) | = | b η a, j ( − k ) | , we see that (( η a, j ) ′ , η a, j ) = P k ∈ Z k | b η a, j ( k ) | = 0 . For convenience we replace j + 1 by 1 /h and a by 1 /q . Then, 0 < h ≤ /
2, 0 < q ≤ h → , and q → . However, to avoid the fussiness of notations we keep the former name for the function η a, j . By (11), k η a, j k = X k ∈ Z | b η a, j ( k ) | = exp (cid:18) − hq (cid:19) X k ∈ Z exp (cid:0) − hqk (cid:1) − exp (cid:18) − h (1 − h ) q (cid:19) X k ∈ Z exp (cid:18) − hq − h k (cid:19) . Using (9) twice for α = 1 , β = 0 , γ = 0 , m = 0, b = 2 hq and b = 2 hq/ (1 − h ) , we get k η a, j k = exp (cid:18) − hq (cid:19) r π hq − exp (cid:18) − hq (1 − h ) (cid:19) s π (1 − h )2 hq + (cid:16) e C ( h, q ) + e C ( h/ (1 − h ) , q ) (cid:17) O (1) , (12)as hq → +0 , where C ( h, q ) = − h/q − ( π − ε ) / (2 hq ) . Similarly, to estimate the quantities k ( η a, j ) ′ k , by (11), we write14 π k ( η a, j ) ′ k = X k ∈ Z k | b η a, j ( k ) | = exp (cid:18) − hq (cid:19) X k ∈ Z k exp (cid:0) − hqk (cid:1) − exp (cid:18) − hq (1 − h ) (cid:19) X k ∈ Z k exp (cid:18) − hqk − h (cid:19) . Using (9) twice for α = 1 , β = 0 , γ = 0 , m = 1, b = 2 hq and b = 2 hq/ (1 − h ) , we get14 π k ( η a, j ) ′ k = 12 exp (cid:18) − hq (cid:19) r π (2 hq ) −
12 exp (cid:18) − hq (1 − h ) (cid:19) s π (1 − h ) (2 hq ) + (cid:16) e C ( h, q ) + e C ( h/ (1 − h ) , q ) (cid:17) O (1) , as hq → +0 , where C ( h, q ) is defined after formula (12). So, recalling (( η a, j ) ′ , η a, j ) = 0 , by Definition 2, weget the following asymptotic form for the frequency variance:14 π k ( η a, j ) ′ k k η a, j k ∼ hq as h → π k ( η a, j ) ′ k k η a, j k ∼ hq as q → . (13)To estimate the first trigonometric moment τ ( η a, j ) (see Definition 2), by (11), we obtain12 π | τ ( η a, j ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z b η a, j ( k ) b η a, j ( k + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − hq X k ∈ Z s(cid:18) − exp (cid:18) − h ( k q + 1)(1 − h ) q (cid:19)(cid:19) (cid:18) − exp (cid:18) − h ( q ( k + 1) + 1)(1 − h ) q (cid:19)(cid:19) e − hq (2 k +2 k +1) . | τ ( η a, j ) | :12 π | τ ( η a, j ) | = e − hq − hq − h r π q (cid:18) √ h + (1 − h )(16 − q ) − q q (1 − h ) √ h (cid:19) + O ( h | ln h | ) for a fixed q ≤ h → , (14)12 π | τ ( η a, j ) | = e − hq (cid:18) e − hq r π hq + O (cid:16) √ q e − h q (1 − h ) (cid:17)(cid:19) for a fixed h ≤ / q → . (15)Let us prove the estimate (14). Put d := 2 h − h , v( k ) := qk + 1 q , s ( k ) := 2 k + 2 k + 1 . (16)Thus, the first trigonometric moment is rewritten as follows:12 π | τ ( η a, j ) | = e − hq X k ∈ Z q (1 − e − d v( k ) )(1 − e − d v( k +1) )e − hqs ( k ) . (17)Using the Taylor formula for the function f ( d ) = p (1 − e − d v( k ) )(1 − e − d v( k +1) ) in the neighborhood of d = 0,we get f ( d ) = p v( k )v( k + 1) d − p v( k )v( k + 1)(v( k ) + v( k + 1)) d + f ′′′ ( ¯ d )6 d , and f ′′′ ( d ) = N − M − (cid:18) ν N (1 − M )(3+ M ) − µνM N (1 − M )(1 − N ) (cid:16) µM + νM +( ν + µ ) M N (cid:17) + ν M (1 − N )(3+ M ) (cid:19) , where N := 1 − e − d v( k ) , M := 1 − e − d v( k +1) , ν := v( k ) , µ := v( k + 1) . We have | f ′′′ ( ¯ d ) | d = O ( s ( k ) h ) . Indeed, f ′′′ is a decreasing function on 0 < d <
1. Collecting summands appropriately, one can check that f ′′ is a concave function on 0 < d <
1. So, | f ′′′ ( d ) | ≤ lim d → f ′′′ ( d ) = 1 / √ µν (5 µ + 6 µν + 5 ν ) . It remains tonote that lim k →∞ lim d → f ′′′ ( d ) /s ( k ) = q / α = 1 , β = 0 , γ = 0 , m = 3 , b = hq , we have for the reminder of (17)e − hq X k ∈ Z f ′′′ ( ¯ d ) d e − hqs ( k ) = O h X k ∈ Z s ( k )e − hqs ( k ) ! = h O (cid:18) h − / + e − π − ε hq O (1) (cid:19) = O ( h / ) , as h → . Therefore, (17) takes the form12 π | τ ( η a, j ) | = e − hq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z p v( k )v( k + 1) (cid:18) d −
14 (v( k ) + v( k + 1)) d (cid:19) e − hqs ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( h / ) . With u := 1 /k we define the function g by g ( u ) := 1 k p v( k )v( k + 1) = 1 k s(cid:18) qk + 1 q (cid:19) (cid:18) q ( k + 1) + 1 q (cid:19) = s(cid:18) q + u q (cid:19) (cid:18) q ( u + 1) + u q (cid:19) . Using the Taylor formula for g ( u ) in the neighborhood of u = 0, we obtain g ( u ) = q + qu + q u + g ′′′ (¯ u )6 u , where g ′′′ ( u ) = 12 g ( u ) 6 u (2 /q + 2 q ) + 6(2 u/q + 2(1 + u ) q ) q − g ( u ) (cid:0) ( u /q + q )(2 /q + 2 q )+ 4 u (2 u/q + 2(1 + u ) q ) + 4( u /q + (1 + u ) q ) q (cid:19) (cid:18) ( u /q + q )(2 u/q + 2(1 + u ) q ) + 2 u ( u /q + (1 + u ) q ) q (cid:19) + 38 g ( u ) (cid:18) ( u /q + q )(2 u/q + 2(1 + u ) q ) + 2 u ( u /q + (1 + u ) q ) q (cid:19) =: 1 g ( u ) P ( u ) + 1 g ( u ) P ( u ) + 1 g ( u ) P ( u ) . Suppose k = 0 , then − ≤ u ≤ . For a fixed 0 < q ≤
1, the value | g ′′′ (¯ u ) | is bounded. Indeed, since q + u /q ≥ q and q ( u + 1) + u /q ≥ q/ ( q + 1) , then 0 < /g ( u ) ≤ p q + 1 /q, and the polynomials P ( u ) , P ( u ) , P ( u ) in u are bounded on [ − , . So, g ′′′ (¯ u ) u = u O (1) . Therefore, we have12 π | τ ( η a, j ) | = e − hq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Z ,k =0 (cid:18) k q + kq + 1 q + O (1) k (cid:19) (cid:18) d −
14 (v( k ) + v( k + 1)) d (cid:19) e − hqs ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( h ) .
6n the latter formula, we omit the summand for k = 0 which equals to e − hq d p /q ( q + 1 /q ) (1 − / /q + q ) d ) e − hq = O ( h ) . Recalling (16), we estimate the coefficient of O (1) in the latter series A := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 k (cid:18) d −
14 (v( k ) + v( k + 1)) d (cid:19) e − hqs ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d ∞ X k =1 k e − hqs ( k ) + qd ∞ X k =1 s ( k ) k e − hqs ( k ) + d q ∞ X k =1 k e − hqs ( k ) . The first series is the main term as h → . Indeed, since P ∞ k =1 s ( k ) k e − hqs ( k ) < P k ∈ Z s ( k )e − hqs ( k ) and P ∞ k =1 1 k e − hqs ( k ) < P k ∈ Z e − hqs ( k ) , then applying (9) for α = 2 , β = 2 , γ = 1 , b = hq, m = 0 , and m = 1 we get d P k ∈ Z s ( k )e − hqs ( k ) ∼ h / , d P k ∈ Z e − hqs ( k ) ∼ h / , as h → . Hence, A ≤ C d ∞ X k =1 k e − hqk = C d e − hq + ∞ X k =2 k e − hqk ! ≤ C d (cid:18) e − hq + Z ∞ x e − hqx d x (cid:19) = C d (cid:18) e − hq + Z ∞√ hq x e − x d x (cid:19) = C d (cid:18) e − hq − e − h q ln( hq ) + Z ∞√ hq x e − x ln x d x (cid:19) = O ( h | ln h | ) . Similarly, one can estimate P k< . Finally, recalling (16), we have12 π | τ ( η a, j ) | = e − hq × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − d q X k ∈ Z s ( k )e − hqs ( k ) + d q − d + dq ) X k ∈ Z s ( k )e − hqs ( k ) + d (2 − q )(2 q − d )4 q X k ∈ Z e − hqs ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( h | ln h | ) . (18)Here, we return the summand for k = 0, since it equals to d/q (1 − ( q + 1 /q ) d/ − hq = O ( h ) . To obtain (14),it remains to substitute (9) for α = 2 , β = 2 , γ = 1 , b = hq, m = 0 , , π | τ ( η a, j ) | = π q h (cid:18) q − q q h + O ( h / | ln h | ) (cid:19) and k η a, j k = π q h (cid:18) q q h + O ( h ) (cid:19) . Finally, substituting the last expressions and (13) in Definition 2 and calculating the limit we get lim j →∞ U C ( η a, j ) = . Let us prove (15). We start with (17). Using the mean value theorem for f ( x ) = √ − x, x = 0, we obtain12 π | τ ( η a, j ) | = e − hq X k ∈ Z (cid:16) − C ( q, k )e − d v( k ) (cid:17) (cid:16) − C ( q, k + 1)e − d v( k +1) (cid:17) e − hqs ( k ) = e − hq X k ∈ Z e − hqs ( k ) − X k ∈ Z C ( q, k )e − d v( k ) e − hqs ( k ) − X k ∈ Z C ( q, k + 1)e − d v( k +1) e − hqs ( k ) + X k ∈ Z C ( q, k ) C ( q, k + 1)e − d (v( k )+v( k +1)) e − hqs ( k ) ! , where C ( q, k ) := 1 / (2 p − c ( q, k )) , < c ( q, k ) < e − d v( k ) . The first series is the main term as q → . Indeed, C ( q, k ) is bounded (for example, since 0 < e − d v( k ) < / < h ≤ / , and 0 < q < (1 − h )(2 h ) − log 2 , then 1 / < C ( q, k ) < √ / S := √ / P k ∈ Z e − d v( k ) e − hqs ( k ) , S := √ / P k ∈ Z e − d v( k +1) e − hqs ( k ) , and S := √ / P k ∈ Z e − d (v( k )+v( k +1)) e − hqs ( k ) respectively. Using (9) for an appropriate α, β, γ, b, and m = 0 , we see that S n = O ( √ q e − dq ) for n = 2 , , . Toobtain (15) it remains to apply (9) (for α = 2, β = 2 , γ = 1 , b = hq, m = 0) to the first series P k ∈ Z e − hqs ( k ) . Finally, substituting (12), (13), and (15) in Definition 2 and calculating the limit we obtain lim a →∞ U C ( η a, j ) = . This completes the proof of Lemma 3. (cid:3)
Proof of Theorem 4.
1. By Theorem 1, the family Ψ a := n , ψ aj,k : j = 0 , , . . . , k = 0 , . . . , j − o (see (8))forms a Parseval wavelet frame for L (0 ,
1) for a fixed a > . Indeed, using definition (7) and the elementaryidentity j − ( j − − = ( j − − − j − , we get b ξ aj ( k ) = J − Y r = j +1 ν r,ak ∞ Y r = J ν r,ak = J − Y r = j +1 ν r,ak exp (cid:18) − k + a ( J − a (cid:19) , j ≤ J − , ∞ Y r = j +1 exp (cid:18) − k + a r ( r − a (cid:19) = exp (cid:18) − k + a ja (cid:19) , j > J − , (19)7here J = ⌊ log ( | k − / | + 1 /
2) + 3 ⌋ . Therefore, the coefficients b ξ aj ( k ) are well-defined. Then a straightforwardcalculation shows that conditions (1)-(4) hold.2. According to Remark 1, let us check lim j →∞ sup a> U C ( ξ aj ) = 1 / a →∞ sup j ∈ N U C ( ξ aj ) = 1 / ξ a, j ( x ) := X k ∈ Z e − k a ja e π i kx = e − aj X k ∈ Z e − k ja e π i kx . (20)Since the U C is homogeneous, it follows that
U C ( ξ a, j ) = U C (cid:16)n e − k ja o(cid:17) . It is known (see [22]) thatlim j →∞ U C (cid:0)(cid:8) e − k j (cid:9)(cid:1) = 1 / . Substituting ja for j to the last equality and swapping j and a we immediatelyget lim j →∞ sup a> U C ( ξ a, j ) = 12 , lim a →∞ sup j ∈ N U C ( ξ a, j ) = 12 . So, taking into account the continuity of the UC (see Lemma 1), it remains to prove that lim j →∞ k ξ aj − ξ a, j k W =0 uniformly on a > , and lim a →∞ k ξ aj − ξ a, j k W = 0 uniformly on j ∈ N . Applying the elementary observation to c j,k = b ξ aj ( k ) − b ξ a, j ( k ) , (namely, if lim j →∞ c j, = 0 and lim j →∞ P k ∈ Z k | c j,k | = 0 , then lim j →∞ P k ∈ Z | c j,k | = 0)we see that it is sufficient to check lim j →∞ k ( ξ aj ) ′ − ( ξ a, j ) ′ k = 0 , and lim a →∞ k ( ξ aj ) ′ − ( ξ a, j ) ′ k = 0 . Using thedefinition of ν r,ak , we get b ξ aj ( k ) = ∞ Y r = j +1 ν r,ak = ∞ Y r = j +1 exp (cid:18) − k + a r ( r − a (cid:19) = exp (cid:18) − k + a ja (cid:19) = b ξ a, j ( k ) (21)for k = − j − + 1 , . . . , j − . Let us consider coefficients b ξ aj ( k ) and b ξ a, j ( k ) for | k − / | + 1 / ≥ j − , that is for j ≤ J − . Denoting ν r,a, k := exp (cid:16) − k + a r ( r − a (cid:17) , recalling (19), and using | ν r,a, k | ≤ , | ν r,ak | ≤
1, we obtain (cid:12)(cid:12)(cid:12)b ξ aj ( k ) − b ξ a, j ( k ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J − Y r = j +1 ν r,ak − J − Y r = j +1 ν r,a, k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − k + a ( J − a (cid:19) ≤ (cid:18) − k + a ( J − a (cid:19) . Applying this estimate, (21), and elementary inequalities ⌊ log ( k + 1) ⌋ + 2 ≤ k / and k + a ≥ a / k / ( a, k ≥ k ( ξ aj ) ′ − ( ξ a, j ) ′ k = X k ∈ Z k (cid:12)(cid:12)(cid:12)b ξ aj ( k ) − b ξ a, j ( k ) (cid:12)(cid:12)(cid:12) ≤ ∞ X k =2 j − k exp (cid:18) − k + a )( ⌊ log ( k + 1) ⌋ + 2) a (cid:19) ≤ ∞ X k =2 j − k exp (cid:18) − a / k / (cid:19) . The last expression is a remainder of a convergent series. Therefore, it tends to 0 as j → ∞ . Moreover, the lastseries converges uniformly on a > . So, it tends to 0 as a → ∞ . The uniformness on j ∈ N is clear. Thus wehave (5).3. To check (6) we use the same method as above. The functions ψ aj , η aj , η a, j (definitions are given in (8), Re-mark 1, (11)) play the role of ϕ aj , ξ aj and ξ a, j respectively. By (8) and Remark 1, b η aj ( k ) = e π i2 − j − k ν j +1 ,ak +2 j b ξ aj +1 ( k ) . Since ν j +1 ,ak +2 j = p − exp ( − k + a ) / ( j ( j + 1) a )) , b ξ aj +1 ( k ) = b ξ a, j +1 ( k ) = exp( − ( k + a )( j + 1) − a − ) for k = − j − + 1 , . . . , j − , then, recalling (11) we conclude (compare with (21)) b η aj ( k ) = b η a, j ( k ) as k = − j − + 1 , . . . , j − . Using the same arguments as for the scaling sequence in item 2, it can be shown thatlim j →∞ k ( η aj ) ′ − ( η a, j ) ′ k = 0 and lim a →∞ k ( η aj ) ′ − ( η a, j ) ′ k = 0 are fulfilled uniformly on a > j ∈ N respectively. Therefore by Lemma 1, lim j →∞ sup a> | U C ( η aj ) − U C ( η a, j ) | = 0 and lim a →∞ sup j> | U C ( η aj ) − U C ( η a, j ) | = 0 . Hence, to conclude the proof of Theorem 4 it remains to use Lemma 3. (cid:3) a . . .
01 1 .
01 100 1000 j · ·
10 10
U C ( ψ aj ) 1 .
497 1 .
498 1 .
496 1 .
497 0 . . U C ( ψ aj ) for particular a ’s and j ’s.8 ææææææææææææææææææææææææææææææææææææææææà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ììììììììììììììììììììììììììììììììììììììììì
400 600 800 1000 1200 j (a) æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ìììììììììììììììììììììììììììììììì
100 150 200 250 a (b) Figure 1: (a) Values of
U C ( ψ aj ) for fixed a : ”circles”, ”squares”, and ”diamonds” correspond to a = 2, a = 5,and a = 10 respectively.(b) Values of U C ( ψ aj ) for fixed j : ”circles”, ”squares”, and ”diamonds” correspond to j = 5, j = 15, and j = 30respectively. In Theorem 4, we get the optimal UC as j → ∞ for the scaling sequences, but the wavelet sequences have UCsequal to 3 / ψ ∈ L ( R ) be a wavelet function on the real line. Put ψ pj,k ( x ) := 2 j/ P n ∈ Z ψ (2 j ( x + n ) + k ) . The sequence ψ pj,k is said to be a periodic wavelet set generatedby periodization . We get the following Theorem 5
Suppose { j/ ψ (2 j ·− k ) } j,k ∈ Z is a Bessel sequence and (( ψ ) ′ , ψ ) L ( R ) = 0 , then lim j →∞ U C ( ψ pj,k ) ≥ / . Proof.
Under aforementioned restrictions the equality
U C H ( ψ ) ≥ / j →∞ U C ( ψ pj,k ) = U C H ( ψ ) . (cid:3) These arguments motivate a conjecture: if ( ψ ′ j , ψ j ) L (0 , = 0 , then lim j →∞ U C ( ψ j ) ≥ / ψ j ) j . If this is true, the family of Parseval wavelet frames constructed in Theorem 4 has theoptimal UC. To prove the conjecture is a task for future investigation.In conclusion we note that the periodization of a real line wavelet function can not provide a result strongerthan the one in Theorem 4. Namely, suppose f n ∈ L ( R ) , n ∈ N is a sequence such that lim n →∞ U C H ( f n ) =1 / . Using the periodization we define sequences of periodic functions f n,pj ( x ) := P k ∈ Z f n (2 j ( x + k )) andapplying results from [23] we get only lim j →∞ lim n →∞ U C ( f n,pj ) = 1 / . However, it is weaker than the equalitiesof the form (5), (6).
Acknowledgments.
The authors thank Professor M. A. Skopina for valuable discussions.