Orthonormal Wavelets Arising From HDAFs

Transcription

1 Contemporary Mathematics Orthonormal Wavelets Arising From HDAFs Jiansheng Yang, Lixin Shen, Manos Papadakis, Ioannis Kakadiaris, Donald J. Kouri, and David K. Hoffmam Abstract. Hermite Distributed Approximating Functionals HDAFs were introduced by Hoffman and Kouri for the numerical computation of the solutions of certain types of PDEs. These functions and their Fourier transforms are both dominated by Gaussians. In this paper we modify HDAFs in order to construct a family of continuous orthonormal wavelets. These wavelets can have any prescribed number of vanishing moments. The main part of the paper is devoted to the study of the smoothness of these wavelets.. Introduction and Main Results The theory of Distributed Approximating Functionals DAF s was introduced in the work of Hoffman, Kouri, and their collaborators e.g. see [HSK9], [HK93]. DAFs are infinitely differential functions very localized in the spatial domain, which, depending on a certain parameter, form sequences of functions converging to the Dirac s delta function in the sense of distributions. This property made DAFs useful for the computation of numerical solutions of certain types of PDEs. In an effort to understand their efficiency for computational applications, the rigorous study of the mathematical foundations of DAFs was intitiated in [CG99] and continued in [WWH0]. The first DAFs that were created were the Hermite DAFs HDAFs, see [HSK9] of whom their definition we include in the next paragraph. The present paper is primarily devoted to the study of the smoothness of the Modified Hermite DAF orthonormal scaling functions and of their corresponding orthonormal wavelets, which were introduced in [KPS, KPS]. A Hermite DAF HDAF of order,, 3,..., is defined by the following equation: h,σ x πσ e x σ n x 4 n! H n, σ 99 Mathematics Subject Classification. Primary 4C40; Secondary 65T60. Key words and phrases. Hermite Distributed Approximating Functionals, wavelet. This research was supported in part by the following grants: Texas Higher Eduaction Coordinating board ARP , SF DMS , SF CHE-00743, and R.A. Welch Foundation E Additional financial support was also provided by the Texas Learning and Computation Center of the University of Houston. c 0000 copyright holder

2 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA where σ is a positive constant and H n is the Hermite polynomial of degree n. The Hermite polynomial H n can be expressed by Rodrigues Formula H n x n e x d n dx e x see also [AS7]. The Fourier transform of the HDAF is routinely shown to n be. ĥ,σ ξ e ξ σ ξ σ n n, n! where the Fourier transform of an L -function f is defined by fξ : fxe ixξ dx, ξ R. It is clear that lim ĥ,σ ξ and lim σ 0 ĥ,σ ξ for all ξ R. These two properties justify why HDAFs form sequences of functions converging to the Dirac s delta function in the sense of distributions see [CG99, WWH0]. We will refrain from repeating the definition of multiresolution analysis MRA, since it is widely known. We only point out that we use the definition of an MRA given in [HW96] and that an orthonormal scaling function is a square-integrable function φ satisfying, first, a two-scale relation, i.e.. φξ m0 ξ φξ a.e. in R, where m 0 is π-periodic and belongs to L [ π, π; second, the integer translates of φ form an orthonormal system. The closed linear span of this system is the core subspace V 0 of the MRA {V j } j, j Z. The π-periodic function m 0 associated with φ is called the low pass filter associated with φ. The wavelet ψ corresponding to φ is given by ψξ e iξ/ m 0 ξ/ π φξ/. The most widely known and utilized wavelets are these with compact support in the time domain, because the wavelet transforms induced by them are implemented by finite impulse response FIR filter banks. However, wavelets associated with low pass filters, which are not trigonometric polynomials are implemented by infinite impulse response IIR filter banks, and such filterbanks are less popular than their FIR counterparts. However, IIR filters with sharp frequency selectivity may be more desirable for certain types of applications, especially if they are symmetric as well. The low pass filters associated with the scaling functions we study in this paper have both these two properties. ext, we present the construction of the MHDAF orthonormal scaling functions. Let be a non-negative integer and σ > 0. We define m,σ, which we also extend π-periodically over the real line, by the following equation: m,σ ξ : We also impose the following condition: π.3 ĥ,σ ĥ,σ ξ π, π ξ < π, ĥ,σ ξ, π ξ π, ĥ,σ ξ π, π < ξ < π..

3 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 3 The definition of m,σ and.3 readily imply.4 m,σ ξ m,σ ξ π, ξ R. Since, ĥ,σ0 we obviously have m,σ 0 and m,σ ±π 0. It is also easy to see that m,σ is a π-periodic continuous function. On the other hand, ĥ,σ is decreasing in the interval [0,. Therefore inf m,σξ m,σ π/ ξ π/ > 0. If we combine this fact with.4, we obtain that φ, defined by φξ m,σ ξ/ j, ξ R, j is an orthonormal scaling function see [Mal89] or Corollary 4.4 in [HW96], which we call a Modified-HDAF MHDAF scaling function. Apparently m,σ is the low pass filter corresponding to φ. We call m,σ the Modified-HDAF low pass filter of order. Since every m,σ is even and takes on non-negative values only, the Fourier transforms of MHDAF scaling functions are even and non-negative. Thus the MHDAF scaling functions are themselves even in the time domain. MHDAF low pass filters have an infinite number of nonzero Fourier coefficients, and the magnitude of the k-th coefficient is majorized by O k 3 [KPS, KPS]. Thus MHDAF scaling functions are not compactly supported. This poses some technical constraints in the study of their smoothness properties, since it leaves us with one choice only: the study of the decay properties of their Fourier transforms. There is a considerable literature devoted to the study of this subject e.g. [CR95, HE95, V94]. The reader may also find a in [S99] a brief and rather comprehensive account on the various smoothness measures of refinable tempered distributions based on the decay of their Fourier transform along with a generalization of Proposition 3.5 of [CR95] Theorem.. However, the conditions guaranteeing a certain degree of smoothness for scaling or refinable functions are not always easy to check, as the reader will soon realize see also [FS0]. Remarkably enough, Theorem.4 reveals that, the Fourier domain techniques proposed in [CR95] for the study of the smoothness of compactly supported scaling functions primarily Propositions 3.3 and 3.5 can also be applied in the case of the MHDAF scaling functions. The investigation of the smoothness of the MHDAF scaling functions begins with the study of the relationship between σ and, the two parameters that determine an HDAF. Recall that.3 implicitly determines σ in terms of, which is a non-negative integer. The unique σ satisfying.3 will be denoted by σ. The estimation of the smoothness of the MHDAF scaling function is given by Theorem.5. However, before obtaining this result we need several intermediate results, namely Theorems.,. and.4. We prove these theorems in the following sections. Theorem.. Set γ : π σ 8. Then the following are true: γ <.08 for, and γ < for ; γ > 3 for all ;

4 4 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA In order to improve the readability of the paper we introduce the following notation; σ, ĥ,σ and m,σ will be denoted by σ, ĥ and m, respectively. Theorem.. The following are true: For every, the low pass filter m has a zero of order of at each ξ k π, with k Z. If is odd, then m is C at k π, for all k Z; if is even, then m is C at k π, for all k Z. 3 m is C at every kπ π/, k Z. At all other points m is C. For α nβ, where n is a non-negative integer and 0 < β, we define C α to be the set of all bounded functions f which are n times continuously differentiable and their n-th order derivative f n is Hölder continuous of order β, i.e. for every x, h R. f n x h f n x C h β Lemma.3. [HW96] Let f L R and α > 0. Suppose j π j π ˆfξ dξ C αj, j 0,,,.... Then the following are true: If α is not an integer, then f C α, otherwise f C α ɛ, for every 0 < ɛ < α. The previous technical result is Lemma 3. in [HW96]. Using Lemma.3 we obtain that if ˆfξ C ξ α, then f belongs to C α, if α is not an integer; otherwise f C α ɛ, for every ɛ in 0, α. For each 0 we define the function m ξ L ξ : cos ξ/. Theorem. implies that L is well-defined everywhere on R. The following theorem is the key result we need in order to prove Theorem.5. Theorem.4. The following are true: L ξ is an increasing function on the interval [0, π]; L ξl ξ is a decreasing function on the interval [ π 3, π]. From the previous theorem, we have L ξ L π 3 for ξ π 3, L ξl ξ L π 3 π for 3 ξ π. This allows us to apply Propositions 3.3 and 3.5 in [CR95] in order to derive Theorem.5 below, which is the main result of this paper. Although these two results from [CR95] require that L is C, it is enough to use the C continuity of L to prove them. Theorem.5. For every,,... we have 0 φξ C ξ b with b : log L π/3 / log. Therefore, φ C b ɛ R, for every ɛ > 0. Furthemore, lim b. ow, from lim b, we infer that, the smoothness of the Modified HDAF scaling functions increases, as increases.

5 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 5. Estimating σ In this section, we prove Theorem.. We begin with a lemma, which will be frequently used in Sections and 4. The proof of the lemma is straightforward, so it will be omitted. Lemma.. Let a, b and c be arbitrary real numbers such that ac bac > 0 and bc 0. Then a b. a c b a a c. Proof of Theorem.. Then, g γ γ e γ g γ g γ : e γ Define γ n n! γ n n! γ n n!, γ R.! 0, so g is decreasing on [0,. On the other hand,. Therefore, in order to prove the first part of Theorem., it is enough to show g.08 < and g < for every. Likewise, in order to establish the second part of Theorem., it is enough to show g 3 >, for all. By direct computations one can verify that g.08, and g for. I :, 3, 4 are less than, hence γ <.08 and γ < for, 3, 4. Let 5. ow, set { and I : k n n! n n n! k { k On the other hand, it is easy to see > k k }! } k!. > k k k k. Moreover, setting a, b 5 and c n, where n,,..., k, Lemma. implies 5 > k k k k 4 5 > 5 5 >. 5 k k Combining the last two inequalities with the definitions of I and I, we obtain I < I.

6 6 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA ow, using lemma. for a I I I, b I I and c I we derive g I I I I I I I I I I 3 <. The final conclusion comes from the fact that γ is decreasing on the positive half-axis. For all define 3 II : n 3 and II : n. n! n! Then and II 3! 3 II 3! 3 k0 n 3 k 3 3 3!. 4 3! Applying Lemma. again, with a II, b II 4 3! and c II we obtain, g 3 which implies γ > 3. II > 4 II II 4 3 >, 3. The Vanishing Moments of the MHDAF Orthonormal Wavelets In this section we prove Theorem.. Proof of Theorem.. By the symmetry and the periodicity of m ξ, we only have to check the continuity and the order of the zero of m at ξ π. Using.4 and m ξ π m ξ π, which is true for all ξ, we have: m ξ ĥ ξ π in π δ, π δ, where 0 < δ < π. For notational convenience, let us define dξ : ĥ ξ. The conclusion of Theorem. will follow, once we have established the properties of d at the origin. Since, ĥ ξ ĥξ ĥ ξ ĥξe ξ σ n ξ σ n n n! ĥξ σ ξ e ξ σ n ξ σ n n n!,

7 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 7 dξ can be factored as 3. dξ ξ gξ, where gξ ĥ ξ σ e ξσ n ξ σ n n n! Clearly, gξ C. Hence, if is odd, then dξ C, and the order of the zero of m at ξ π is equal to. If is even, then d is only C at 0, but the order of the zero of m at ξ π is still equal to. We ommit the proof of the last item in Theorem., because it is straightforward. 4. The Smoothness of M-HDAF Scaling Function This section begins with the proof of the first part of Theorem.4. Then, we give some technical lemmas, useful only in the proof the second part of Theorem.4. The proofs of the second part of Theorem.4 and of the last assertion of Theorem.5 conclude the present section. Let us first introduce some auxilliary functions. Let γ > 0. Define, γ γ n Q, γ :! n!, Clearly, Q, γ : Q,3 γ : Q,4 γ : γ! Q, γ Q, γ Q,3 γ γ n n! n γ n n!, n γ n n!, Q, γ Q,3 γq, γ. n γ n n ;! n!γ n ; γ Q,γ Q, γ. Lemma 4.. The following are true for every : Q, γ γ ; Q, is decreasing on [0,, and Q, > Q,. 3 Q,4 γ Q,3γ, and Q,4 is increasing on 0, ]. Proof.

8 8 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA Using 4., we have Since Q,γ Q, γ γ n n γ n 3 n γ n n γ γ n! n! n n we conclude that Q, is decreasing on [0,. But, and! k >!! > γ. γ! k for every k. Then, using., we have Q, > Q,. < 0, 3 Eq. 4. implies Q, γ. This inequality and Q,3 γ, for all γ 0, imply Q,4 γ Q, γ On the other hand, it is easy to check Then, we clearly have, Q,4 γ Q,3 γ Q,3 γ Q,3γ. γ! e γ. γ n n! γ Q!,4γ eγ γ n n! γ! eγ γ n n! e γ > 0 γ n n! for every γ 0, ], which implies that Q,4 is an increasing function on the interval 0, ]. Proof of item of Theorem.4. In order to establish that L is increasing on the interval [0, π], it is sufficient to prove L 0 on each one of the intervals [0, π/] and [π/, π]. First let ξ [0, π/]. Then ĥ ξ L ξ cos ξ/. In order to show L ξ 0, for every ξ [0, π/], it is enough to prove ĥ ξ cos ξ/ ĥ ξ cos, ξ/

9 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 9 equivalently, tan ξ/ σ ξ Q,σ ξ /. First, let 4. Then by item of Theorem., we have σ ξ σ π max.08,, 8 so routine calculations imply σ ξ σ Q ξ, ξ 4 tan ξ, for,, 3, 4. On the other hand, if 5, then item of Theorem., gives σ 8 π. Then, σ < 8 π and item of the previous lemma implies σ Q ξ, Q, Q, Q 5,5 < 7. Hence σ ξ Q, σ ξ 6 7π ξ.3ξ π ξ 4 tanξ/, which establishes that, for 5, L is increasing on [0, π/]. ow, let ξ [ π, π]. Define L ξ : L π ξ ĥ ξ cos π ξ ĥ ξ sin ξ/. We must now prove that L is decreasing on [0, π ]. To accomplish this goal it is enough to show ĥ ξ or equivalently, 4.4 But, 4. implies Q, σ ξ σ is true as well: sin ξ/ ĥ ξ sin ξ/ tan ξ/ ξ Q, σ ξ. ξ 4.5 tanξ/ ξ σ ξ 3 4. ow, item of Theorem. implies σ > 8 is enough to prove, so 4.4 is true if the following inequality π tan ξ ξ 6ξ3 3π,. Therefore, to obtain 4.5, it

10 0 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA for every ξ [0, π/4]. So, let Then gξ : tan ξ ξ 6ξ3 3π. g ξ sec ξ 6ξ π tan ξ 6ξ π tan ξ ξ ξ 6 π tan ξ ξ ξ 6 π ξ π 4 0. so, gξ g0 0, for every ξ [0, π/4]. This completes the proof of 4.5 and so of the first part of Theorem.4. The proof of second part of Theorem.4 is more complicated and it requires several intermediate results, which we will state and prove in the sequel. Lemma 4.. Let x 0, π/4]. Then, the following are true: x 3! x 3 tan x x x 3! x 3 x 3. x3 35 ; Proof. Since the proof of item is easy, we will only give the proof of item. We first have to prove 4.7 tan x x 3! x3 6 5! x5, 0 < x π/8. In order to prove the latter inequality we have to use Taylor s expansion of the tangent function [AS7]: tan x n 4 n 4 n B n x n, n! where B n is a Bernoulli number. Since, B n n n! π n p n for n,,.... p we conclude 4 n 4 n B n > 0 n,,.... n! This establishes 4.7. Using 4.7 we obtain x 3! x 3 tan x x 3! x 3 x 3! x 3 6 5! x 5 x 3 0 3! π 8 3! π 8 6 5! π 8 4 x3 35.

11 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 4.8 Proposition 4.3. For every and x 0, π 4 ] we have x Q,3 σ x Q, σ x tan x tan x 4σ x Q, σ x 0. Proof. First, we discuss the case 3. Applying item of Theorem. we obtain σ x γ, for every x [0, π/4], 4 where γ π σ 8. Eqs. 4. and 4. imply σ x Q, γ 4 n σ x, Q,! n! γ 4 n. ow, let. According to Theorem. γ <.08, so σ 8.08 π. Therefore, σ x σ x Q, 0 9, Q, ow, applying 4.3 we get σ x 4.9 Q,3 Furthermore, item of Lemma 4. gives us 4.0 σx Q, σ x x π. Using the definition of Q,4 one can easily verify π x x Q,3 σ x Q, σ x tan x x tan x x xq, σ x Q,4 σ x x Using the Taylor series expansion of the tangent and the fact that for 0 x < π/ we can easily see tan x x 3! x 3. Combining the previous inequality with item of Lemma 4. we conclude 4. x tan x x 3. On the other hand, 4.0 implies 4.3 x xq, σ x 4.3x 3π. Item 3 of Lemma 4. and 4.9 finally imply 4.4 Q,4 σ x 8. x 7 4x π..

12 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA Combining inequalities 4., 4.3, 4.4 and 4. we obtain x Q,3 σ x Q, σ x tan x x x π x π x x π. Hence, the left-hand-side of 4.8, which we denote by LHS, satisfies LHS tan x x Q, σ x 4x π, so 4.8 will be valid, if the right-hand-side of previous inequality is non-negative for every x 0, π 4 ]. Let us first, assume x 0, π 6 ]. Since σ 8.08 π and Q, is decreasing we obtain Q, σ x Q, > 3. Therefore, tan x x x Q, σ x π tan x x x 3 π x 8x π 0; ow, let x [ π 6, π 4 ]. Arguing as in the previous case we obtain Q,σ x Q, Combining these inequalities with tan x x 3 7x 6, we conclude tan x x x Q, σ x π 7x x 5/7 π 0. The cases and 3 are similar to the case, so we will ommit their proofs. 4.5 ow, let 4. The item of Theorem. implies σ x γ 4 4, for every x [0, π/4], where γ π σ 8. Then, using the previous inequalities and 4. once again, we get σ x Q, 4 n Applying the definition of Q, and using 4.5, we have σ x σ x Q, σ x n! n! ow, observe n n 4 n > 3 n σ x n n n 4 n. n. 4 n Letting n,, 3 and a, b, c in Lemma. be equal to n, 4, n, respectively we obtain n/ > 4 n/4. So Therefore, n 4 n > 4 4 n, n,, 3. σ x Q, σ x n 6. n σ x

13 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 3 Thus, using also 4.3, we conclude σ x 4.6 Q,3 σ σ x π x4 48 x4 7. ext, we will establish the following two inequalities, which clearly imply and 4.8 x Q,3 σ x Q, σ x tan x xq, σ x x 3! x 3 xq, σ x x 3! x 3 4σ tan x x Q, σ x 0. We begin with the proof of 4.7. The definition of Q,4 readily implies xq, σ x x Q,3 σ x Q, σ x Q σ,4 x x But, Q,4 σ x Q,3 σ x, x x due to item 3 of Lemma 4.. Combining the previous inequality with 4.6 we get 4.9 xq, σ x x Q,3 σ x Q, σ x x3 44, which together with item of Lemma 4. imply 4.7. Let us now prove 4.8. Item of lemma 4. and item of 4. imply xq, σ x x 3! x 3 xq, σ x x x x 3! x 3 x 3 σ x x 3 4x π. Using the fact that γ < and the previous inequality, we obtain xq, σ x x 3! x 3 tan x x 3 4σ tan x x Q, σ x 8 Q, σ x 4x π, for all 0 x π/4. So, the conclusion of proposition 4.3 will be established once we have that the RHS of previous inequality is non-negative, for all 0 x π/4 and 4. In order to prove this fact, let us first assume 0 x π 6. Using the fact that Q, is decreasing we derive 4 4 Q, σ x 4! Q, 9 4 n! 6 > 8. 9 n.

14 4 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA Thus, tan x x 8 3 4x Q, σ x π x 8x π 0. ow, let π 6 x π 4. Since, tan x x 3 7x 6, in order to prove 4.0 tan x x 8 3 4x Q, σ x π 0, it is enough to establish x Q, σ x π 7x 6 for all 4 and x π 6, π 4 ]. Since for every x π 6, π 4 ] we have σ x, the fact that Q, is decreasing, implies Q, σ x Q,. Direct calculations show that for 4, 5, 6, 7, 8, 9, Q, is greater than 3., 3.5, 3.77, 4, 4.4, and 4.45 respectively. This establishes 4. for 4, 5, 6, 7, 8, 9 and x π 6, π 4 ]. ow, let 0. Using item of Lemma 4. we have Q, σ x Q, Q 0, 0 > 4.65, which implies 8 4x Q, σ x π 8 4x 4.65 π 7x 6. Therefore, 4.0, and thus, 4.8 hold for every x π 6, π 4 ] and 4. This concludes the proof of the Proposition 4.3. Before stating the next lemma we need to introduce some notation which will be helpful in its proof. If x is an arbitrary real number, we denote by x and x the greatest integer less than or equal to x and the smallest integer greater than or equal to x, respectively. Lemma 4.4. The function gy y is increasing on the interval 0, ]. Q, y Before we give the proof of Lemma 4.4, we first show the plots of the function g for, 0, and 00, respectively, in Figure. Proof of Lemma 4.4. g y where 4. Iy : n!y n n!!y n n! The first derivative of g is y!y n n! Iy, y y n!y n!y n. n! n!

15 ORTHOORMAL WAVELETS ARISIG FROM HDAFS Figure. The plots of the function g in Lemma 4.4 from left to right with, 0, and 00, respectively. Our goal will be to show Iy 0 for all y 0, ]. First, note that the series involved in the definition of I are bounded by geometric series, because 0 < y. This implies that the series in the right-hand side of eq. 4. converge uniformly. ow,!y n n n!!y n n!!y n n n!!y n n n! n y n yn n l 3 y [ n y 3 y3 n4 y 3 k k ik k!y l l l! 34 y3 l y i k ik l!y l l! n l ynl n4 n yn n l l 34 3 k i i k i k y k. ] y n y 3 y k

16 6 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA This expansion of the term!y n n! now yields Iy 3 y y y n 3 n n n n4 n y n 3 n k k ik k k k ik i k i y k i y k. k i Using 0 < y, one can easily verify the following inequalities y i i y k i k i y k k y i i y k i k i y i 3 i ; y k 3 k ; y i 3 i. The previous inequalities imply k ik k ik i k i k y k y i i y k i k i y k k k ik y i 3 i y k 3 k.

17 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 7 Using again 0 < y we have, k ik k ik i y i k i < y k k i ik y i 3 i. Hence k k ik i k k k k k ik y k k i y i 3 i y k 3 k y k 3 k i3 i k i y i 3 i y k 3 k i yi 3 i i3 y 3 k yk 3 k k3 and k k3 k ik i y k k i k yk 3 k.

18 8 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA By adding the last two inequalities we obtain k k ik i k k k ik y 3 k3 y k i k k yk 3 k y 3 k y k 3 k k3 y 3 k y k 3 k k k i y k k i y 3 5y 3 4 7y n y n 3 n. Furthermore, it can easily be verified that n4 n 3 n n n n y n 3 n 4n y n 3 n n y n 3 n. n4 n4 n4 Combining the previous inequalities we readily obtain so, g y 0, for all y 0, ]. Iy 0, y 0, ], Lemma 4.5. For every x [ π 4, π 3 ], define h x a x ax π x aπ x. Then, h x is an increasing function on the interval [ 4 3π, 4 π ].

19 Proof. ORTHOORMAL WAVELETS ARISIG FROM HDAFS 9 Routine calculations yield h x xa ax π x aπ x xaπ xx π x π x x a xπ xx 3 π x 3 ax aπ x π x x a xπ xx 3 π x 3 ax aπ x. First, observe that if x [0, π π 3 ], then π x x π/3, and if x [, π /3 3 ], then x 3 π x 3 0, and thus π x x a xπ xx 3 π x 3 > 0. So, regardless of the values of a, h xa is positive for every x [, π /3 3 ]. ow, assume x [ π 4, π ]. Then, x 3 π x 3 4 < 0, so, if a [ /3 3π, 4 π ], then π x x a xπ xx 3 π x 3 ωx, where ωx : π x x 6 π xπ xx 3 π x 3. The proof of the 4 lemma will be completed, once we establish ωx > 0, for every x in [ π 4, π. /3 First, note that ωπ/4 > 0. On the other hand, ω x 8 π [x x 5 ]. But the quadratic polynomial x x 5 is decreasing on,, thus ω restricted on [ π 4, π π π ] attains its minimum value at x. Since, <, we /3 /3 /3 obtain ω π 6 > ω 5 /3 π > 0. This establishes that ω is increasing in the interval [ π 4, π ], and since ωπ/4 > 0, /3 we finally conclude that h xa > 0, for all x [ π 4, π ] and a [ /3 3 4 π, 4 π ]. Thus, h x is an increasing function on the interval [ 3 4 π, 4 π ]. and Lemma 4.6. Let py y 3 sin y tan y sin y tan y y 3 sin y tan y qy sin y tan y y 3. Then, for every y [ π 8, π 4 ] we have py 0 and that q is increasing. Proof. Obviously, It is also easy to verify py qy sin y tan y. q y y sin y y tan y 3 sin y tan y y 4 tan yy cos y y 3 sin y y 4. ow, let ry y cos y y 3 sin y. π

20 0 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA Then, r y cos y 4y cos y sin y 3 cos y cos y y sin y, r y 4 sin y sin y 4y cos y cos ytan y y. Clearly, r y 0 for all y [0, π 4 ] and r0 r 0 0. Thus, r y > 0 and so ry > 0, for every y [0, π 4 ]. Therefore, q y 0, y 0, π ]. 4 Since lim y 0 qy 0 we have qy > 0 for every y [0, π 4 ], so py > 0 for all y [ π 8, π 4 ]. Lemma 4.7. The function 4x Dx x tan x 3π 4x 3π is non-negative on the interval x [ π 4, π 3 ]. 4π x 3π 4π x 3π π x tan π x Proof. Elementary computations give π π D > 0.03, D > In view of these numerical results, the conclusion the lemma will be established, once we show D x 0 for x [ π 4, π 3 ], because, then D is concave downward. Routine calculations give and 4 D 3π 4x 4 x 3π 3π 4π x 4 3π 4x 3π 4π x 3π x 4 sin x 4 π x 4 sin π x D x 4 3π x3 4x 3π 4x 3π 3 8 x 3 3 9π x3 4x 4 3π 4x 4 4 sin x tan x 3π π π x3 4π x x 3 sin x tan x 4π x 3π 3 3π 8 π x 3 4 sin π x 3 9π π x3 4π x 4 3π 4π x 3π 3 π x 3 sin π x tan π x tan π x For π 4 x π 3, we have π 3 π x π. A routine first-order derivative test shows that ay : y3 4y 3π is increasing on [ π 4, π ]. Applying this result both for.

21 ORTHOORMAL WAVELETS ARISIG FROM HDAFS the restriction of a on the interval [ π 4, π ] and the composite function x aπ x, where π 4 x π 3, we obtain, 3 9π x3 4x 4 3π 3 9π π x3 4π x 8 4 3π 4x 3π 3 4π x and 3 π 9π π π < π 9 π Moreover, Lemma 4.6, implies π x 3 sin π x 3π 3 3 π 9π π π 4 3π tan π x π 4 3 sin x 4 x tan x sin π 3 sin x tan x x 6 tan π sin x tan x π sin π 8 tan π < Combining all the previous inequalities we conclude D x < Eq. 3. in the previous section implies that L is continuous on [0, π]. Therefore, in order to prove Theorem.4 it is enough to restrict our discussion on the interval 0, π. Proof of item of Theorem.4. Since L is even and L ξl ξ L ξl π ξ L ξl π ξ. Changing the variable from π ξ to x, we assert that L ξl ξ is decreasing on [ π 3, π] if and only if L π xl x is increasing on [0, π 3 ], equivalently ln L π xl x 0, for every 0 x π 3. Since, π 3 π x π for every x [0, π 3 ], we get L π x ĥ,σ x sin x. We will give the proof of above inequality on the intervals 0, π 4 and [ π 4, π 3 ], separately. First, let x 0, π 4. Then, 0 < x < π. Therefore, L x ĥ,σx cos x. So, if x 0, π 4, routine calculations give [ln L π xl x] x Q,3 σ x Q, σ x tan x tan x 4σ x Q, σ x.

22 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA A direct application of Proposition 4.3, implies ln L π xl x 0. Therefore, L π xl x is increasing on [0, π 4 ]. We now focus on x [ π 4, π 3. Obviously, all such x satisfy π x < π 3. Therefore, Thus, L xl π x ĥ,σ π x sin π x ĥ,σ x sin x. ln L π xl x x Q,3 σ x Q, σ x tan x π x Q,3 σ π x Q, σ π x We denote the right-hand-side of above equation as I. Using the definition of Q,4, we rewrite I: I I I I 3, where I I π x Q,4 σ π x x Q,4 σ x xq, σ x π xq, σ π x, I 3 tan x tan π x ow, observe that for every x [ π 4, π 3 ] we have, and thus, 4.3 x π x π/, x π x.., tan π x Moreover, due to Theorem., we obtain σ x < γ < and σ π x < γ <, for all and π 4 x π 3 ; so, by 4.3 and item 3 of Lemma 4. we conclude I 0. An elementary calculation establishes I xq, σ x x σ x x σ x π xq, σ π x π x σ π x π x σ π x. Using g defined in Lemma 4.4, we obtain σ I π x g π x x σ g x x σ x π x σ π x..

23 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 3 Combining σ x < and σ π x <, 4.3 and the fact that g is increasing on [0,, for all, 4.4 I x σ x π x σ π x. Using once again γ <, for every, and item of Theorem. we have 4 3π σ 4 π,. Setting a σ and then applying Lemma 4.5, inequality 4.4 implies, I x 4x 3π π x 4π x 3π. Combining the previous inequality with I 0, we have I x 4x 3π tan x x tan x tan π x 4x 3π 4x 3π for π 4 x π 3. Finally, Lemma 4.7 readily implies, π x 4π x 4π x 3π 4π x 3π I 0, which completes the proof of second part of Theorem.4. 3π π x tan π x, We will conclude the present section with the proof of Theorem.5. Proof of Theorem.5. According to the remarks preceding the statement of Theorem.5 the only conclusion that remains to be proved is lim b. The definition of b implies π 4.5 b log ĥ,σ, 3 for every. In order to establish lim b it suffices to prove 4.6 lim ĥ,σ π/3 Using g defined in the proof of Theorem. we infer π ĥ,σ g 3 9 γ, where γ is defined so that ĥ,σ π g γ. Recall that g is decreasing refer again to the proof of Theorem.. By Theorem. γ < for all. Thus, 4 4 g 9 < g 9 γ <.

24 4 J. YAG, L. SHE, M. PAPADAKIS, I. KAKADIARIS, D.J. KOURI, AD D.K. HOFFMA In order to prove 4.6, it is enough to show lim g 4 9. In fact, a more general result is true: lim g x, for 0 < x <. Let us now establish this result. Fix 0 < x <. Using the definition of g we have g x e x x n n x x e! On the other hand the series k n! k x k k series k xk, which converges since 0 < x <. Therefore, x k. k x x 0 g x < e! x. is dominated by the geometric Using Stirling s formula n! > e n n n, for all n for n, we now obtain which finally gives x x e! x < e x x e x, 0 g x < xe x ex x. Observing that the function x xe x is increasing on [0, ] we have xe x <, for every 0 < x <, which establishes lim g x Acknowledgments This research was supported in part by the following grants: Texas Higher Eduaction Coordinating board ARP , SF DMS , SF CHE and R.A. Welch Foundation E Partial finacial support was also provided by the Texas Learning and Computation Center of the University of Houston and by a University of Houston GEAR 00 grant. We finally wish to express our most sincere thanks to Professor Q. Sun, currently with the University of Houston s Mathematics Department, for his valuable suggestions that helped us to improve the original manuscript. References [AS7] M. Abramowitz and I.A. Stegun Eds., Handbook of Mathematical Functions, Dover Publications Inc., Y 97. [CG99] C. Chandler and A. G. Gibson, Uniform approximation of functions with discrete approximation functionals, J. Approximation Theory , [CR95] A. Cohen and R. D. Ryan, Wavelets and Multirate Signal Processing, Chapman & Hall, 995. [FS0] A. Fan, Q. Sun, Regularity of Butterworth refinable functions, Asian J. Math. 53, pp , 00. [HK93] D. K. Hoffman and D. J. Kouri, Distributed approximating function theory for an arbitrary number of particles in a coordinate system-independent formalism, J. Phys. Chem , [HSK9] D. K. Hoffman,. ayar, O. A. Sharafeddin, and D. J. Kouri, On an analytic banded approximation for the discretized free propagator, J. Phys. Chem ,

25 ORTHOORMAL WAVELETS ARISIG FROM HDAFS 5 [HW96] E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, 996. [HE95], L. Herve, Construction et regularite des fonctions d echelle, SIAM J. Math. Anal., 6, pp , 995. [KPS] I.A. Kakadiaris, M. Papadakis, L. Shen, D.J. Kouri, and D.J. Hoffman, g-hdaf multiresolution deformable models for shape modeling and reconstruction, British Machine Vision Conference Cardiff, UK, Sep 00. [KPS], m-hdaf multiresolution deformable models, The 4th International Conference on Digital Signal Processing Santorini, Greece, Jul 00. [Mal89] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of l r, Trans. Amer. Math. Soc , [S99] Q. Sun, Sobolev exponent estimates and asymptotic regularity of the M-band Daubechies scaling functions, Constr. Approx., 5, pp , 999. [V94] L.F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal., 5, pp , 994. [WWH0] G. Wei, H. Wang, D. K. Hoffman, D.J. Kouri, I. A. Kakadiaris, and M. Papadakis, On the mathematical properties of distributed approximating functionals, J. Phys. Chem , no., School of Mathematical Sciences, Peking University, Beijing 0087, P. R. China address: yjs@math.pku.edu.cn Department of Mathematics, West Virginia University, Morgantown, WV address: lshen@math.wvu.edu Department of Mathematics, University of Houston, Houston, TX address: mpapadak@math.uh.edu Department of Computer Science, University of Houston, Houston, TX address: ioannisk@uh.edu Department of Physics, University of Houston, Houston, TX address: Kouri@uh.edu Department of Chemistry, Iowa State University, Ames, Iowa 500 address: hoffman@ameslab.edu