Interpolatory periodic scaling functions and wavelets

INTERPOLATORY PERIODIC SCALING FUNCTIONS AND WAVELETS KOK CHI WEE (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2004 Acknowledgements It has been quite a tough year taking the Accelerated Master’s Degree Programme Being the last scholar on this programme is an honour to me but at the same time, rather pressurizing Full of confidence in the beginning, I started to lose faith in completing the degree as months passed without any fruitful research There was no anxiety initially as my priority was placed on the coursework and teaching assignment Then came the December holidays when much of my time was spent either experimenting cake recipes or dozing off from reading research material It was a period when I simply was not in the mood for research This was followed by another semester of coursework, teaching assignment and zero amount of research this time round It was not until the recent three months of intense research and hard work when almost all the stuff in this thesis was created I attribute this feat to the following people First of all, I express my gratitude to my supervisor, Associate Professor Goh Say Song He trusted that I would eventually come up with something for this thesis when I conveyed to him my lack of motivation Without his encouragement and help, I would have quitted long ago And of course, this thesis is ascribable to his guidance ii Acknowledgements iii Next, I would like to thank my instructors and fellow classmates of the culinary classes I have attended during this arduous period Their joyous company lightened my tense mood Not forgetting my friends, I would like to thank Weiguang, Minghui, Jinfeng, Jeffrey, Jiayi, Fengying, Baoling and Philip whom I played badminton with almost every week The game sessions with them helped me vent my frustrations Last but not least, I would like to show my appreciation to my family members for their utmost care and concern that pulled me through the difficulties My thanks goes to — my sisters for sharing their titbits, comics and teddy bears; my dad for all the little things in life; and my mom for keeping my room clean and cooking the homely meals Chi Wee July 2004 Contents Acknowledgements ii Summary vi Periodic Scaling Functions and Wavelets 1.1 Preliminaries 1.2 Multiresolution of L2 [0, 2π) and Polyphase Splines 1.3 Periodic Wavelets from Polyphase Splines 12 Interpolatory Scaling Functions and Wavelets 31 2.1 Introduction 31 2.2 Interpolatory Properties 33 2.3 Existence of Interpolatory Wavelets 41 iv Contents v Examples of Interpolatory Trigonometric Scaling Functions and Wavelets 59 3.1 Introduction 59 3.2 Generalized de la Vall´ee Poussin Means 60 3.3 A General Family of Trigonometric Scaling Functions 72 Bibliography 88 Summary Wavelets play a major role in many applications like data compression, signal and image processing, and resolution enhancement Many studies have been conducted on wavelets and here we consider interpolatory periodic wavelets, a class of wavelets that provides certain advantage over other wavelets This thesis continues the study of interpolatory periodic wavelets done in the honours thesis [7] Periodic multiresolutions with dilation were considered in the honours thesis and extension is being made by considering a general dilation M that is greater than or equal to Chapter contains a detailed study of periodic multiresolutions with general dilation, together with their corresponding scaling functions and wavelets From this analysis of general periodic scaling functions and wavelets, the extended interpolatory theory is developed in Chapter Examples of interpolatory periodic scaling functions and wavelets are given in Chapter All results in Chapters and are new The paper [4] supplied part of the material in Chapter on which the remaining new material in the chapter is built upon Results in Section 1.2, which is about periodic multiresolutions, are found in [4] Some of these have been formulated or proved differently so as to follow the presentation sequence here In contrast, vi Summary the development of periodic wavelets in Section 1.3 takes a different approach This section contains necessary and sufficient conditions for the existence of various wavelets, some of which are innovated from [4] and are cornerstones of the enhanced interpolatory theory This section also encompasses new material on wavelets that form Riesz bases for L2 [0, 2π) For results which have analogous versions in the honours thesis, their proofs are omitted to avoid duplication whenever there is no significant change in technique Besides the consideration of a general dilation, the generalization of the interpolating height of both scaling functions and wavelets add considerable intricacies to the original interpolatory theory This is reflected in the comprehensive theory developed in Chapter Section 2.2 examines interpolatory properties of scaling functions and wavelets In addition, relations between the interpolating nature and the interpolating height of the scaling functions are established With the aid of the innovated material from Chapter as well as several linear algebra results, the discussion on the existence of interpolatory wavelets in the honours thesis is elevated in Section 2.3 to the much more general setting on hand Section 2.3 also includes a formula for the interpolatory wavelets Modified versions of the de la Vall´ee Poussin means of the Dirichlet kernels for dilation M are created independently and shown to form interpolatory periodic scaling functions in Section 3.2 In Section 3.3, a sharp observation of these modified functions leads to an extensive family of periodic scaling functions, some of which possess interpolation or orthonormality properties It is demonstrated in both sections that corresponding interpolatory periodic wavelets that form orthonormal or Riesz bases for L2 [0, 2π) can be constructed, using the theory in Chapter The examples in Chapter complement the extended interpolatory theory vii Chapter Periodic Scaling Functions and Wavelets Let L2 [0, 2π) be the space of all 2π-periodic square-integrable complex-valued functions with inner product ·, · and norm f, g := 2π · given by 2π fg , f := f, f 1/2 This chapter focuses on the theory of periodic scaling functions and wavelets Based on [4], the discussion here can be viewed as a generalization of the work done in [7] for a single scaling function and wavelet The first section is a brief list of preliminary results This is followed by an analysis of multiresolutions of L2 [0, 2π), analogous to that of [7] but with more technical details to take care of multiple scaling functions This increase in complexity carries over to the last section on wavelets Furthermore, substantial amount of the material in this last section, to be used in subsequent chapters, has no correspondence in [4] or [7] 1.1 Preliminaries The results in this section will be used later and the reader can refer to Chapter of [7] for their proofs, with or without simple modifications 1.1 Preliminaries Proposition 1.1 Suppose {Vk }k is a sequence of nested, finite-dimensional sub- spaces of L2 [0, 2π) and Wk is the orthogonal complement of Vk in Vk+1 for each k (a) Each f ∈ k Vk can be expressed as ∞ f = PV0 f + PW k f , k=0 where PV f denotes the projection of f on the space V (b) Each pair f and g from k Vk satisfies ∞ f, g = PV0 f, PV0 g + PWk f, PWk g k=0 Define fˆ(n) := f, ein· , n ∈ Z, as the Fourier coefficients of any function f ∈ L2 [0, 2π) With this notation, we have the well-known Parseval’s Identity Proposition 1.2 (Parseval’s Identity) For every pair f and g from L2 [0, 2π), fˆ(n)ˆ g (n) f, g = n∈Z In particular, f = n∈Z |fˆ(n)|2 for each f ∈ L2 [0, 2π) Proposition 1.3 Let f ∈ L2 [0, 2π) (a) The function f is real-valued a.e if and only if fˆ(n) = fˆ(−n), n ∈ Z (b) The function f is even a.e if and only if fˆ(n) = fˆ(−n), n ∈ Z Proposition 1.4 Let M be an integer greater than or equal to If a ∈ l2 (Z), then vk,j := p∈Z a(j + M k p)ei(j+M and vˆk,j (n) = k p)· converges for k 0, j = 0, 1, , M k − 1,    a(n) if n ≡ j mod M k ,   0 otherwise 1.2 Multiresolution of L2 [0, 2π) and Polyphase Splines Multiresolution of L2[0, 2π) and 1.2 Polyphase Splines Let r and M be positive integers with M A periodic multiresolution (MR) of L2 [0, 2π) with multiplicity r and dilation M is a sequence of subspaces {Vk }k of L2 [0, 2π) that satisfies the following three conditions MR1 For each k 0, dim Vk = rM k and there exist functions φm k ∈ Vk , m = k 1, 2, , r, such that {Tkl φm k : m = 1, 2, , r, l = 0, 1, , M − 1} is a basis for Vk , where Tkl f := f (· − MR2 For k 2πl ) Mk 0, Vk ⊂ Vk+1 Vk = L2 [0, 2π) MR3 k For k r T 0, the functions φm k , m = 1, 2, , r, and φk := (φk , , φk ) are called scaling functions and scaling vector respectively Let Wk be the orthogonal complement of Vk in Vk+1 for each k It follows that dim Wk = dim Vk+1 − dim Vk = rM k (M − 1) and Wk ⊥ Wk when k = k As to be shown later, analogous to Vk , Wk is generated by {Tkl ψkm : m = 1, 2, , r(M − 1), l = 0, 1, , M k − 1}, where ψkm ∈ Wk , m = 1, 2, , r(M − 1), are called wavelets The term multiwavelets refers to wavelets when r > The subspaces Vk and Wk are known as multiresolution subspace and wavelet subspace respectively Study of periodic MRs and wavelets for r = and M = was done in [3], [6] and [9], with a summarized discussion, which was based on [6], given in Chapter of [7] Here we consider the corresponding theory for more general values of r and M For a function f ∈ L2 [0, 2π), Parseval’s identity implies that f < ∞, that is, fˆ ∈ l2 (Z) Thus for k M k p)ei(j+M k p)· n∈Z 0, j = 0, 1, , M k − 1, converges by Proposition 1.4 Given {φm k : k m L2 [0, 2π), the polyphase splines of these functions, vk,j , k |fˆ(n)|2 = p∈Z fˆ(j + 0, m = 1, 2, , r} ⊂ 0, m = 1, 2, , r, 3.3 A General Family of Trigonometric Scaling Functions trivially For k 1, the orthogonal splines of each   ck ij·  e if  Mk    vk,j = φˆk (j)eij· + φˆk (j − M Nk )ei(j−M Nk )· if M       ck ei(j−M Nk )· if M2 Mk with norm squares   ck2    M  2k vk,j =     |φˆk (j)|2 + |φˆk (j − M Nk )|2 if or if M j M 75 φk are M j Nk − S k , M Nk − S k < j < Nk + S k M Nk + S k , M Nk − , j Nk − S k Nk + S k M Nk − , j Nk − S k < j < M Nk + S k (3.9) Hence vk,j (0) = ck , Mk j = 0, 1, , M Nk − 1, and so by Theorem 2.1, φk interpolates as in (2.1) with interpolating height ck ˆ (lM ) = h ˆ (l), l = 1, 2, , M − 1, vanishes according When k = 0, all of h to (3.7) and thus Case (ii) of Proposition 2.11 is satisfied Alternatively, all of ˆ (lM ), l = 0, 1, , M − 1, are nonnegative and so Case (i) of Proposition 2.11 h is satisfied Since j = is the only value to check, (2.15) holds for k = and interpolatory wavelets ψ0m , m = 1, 2, , M − 1, of the form (2.4) exist The argument for the existence of interpolatory wavelets when k is broken down into three cases Case j M Nk − S k Since j + Mk M k = M Nk = M M Nk + Nk 2 M Nk + S k and j + (M − 1)M k M Nk − Sk + (M − 1)M k M = M Nk+1 − M Nk + Nk − S k 3.3 A General Family of Trigonometric Scaling Functions 76 M Nk − S k , = M Nk+1 − ˆ k+1 (j + lM k ) = for l = 1, 2, , M − 1, from (3.8) Hence Case (ii) of we have h Proposition 2.11 is satisfied by each j in this range As an alternative, M Sk − M Nk + ˆ k+1 (j) = φˆk (j)/φˆk+1 (j) = ck M/ck+1 Nk+1 − Sk+1 implies that h 0, and thus verifying Case (i) of Proposition 2.11 for these values of j Case M Nk − S k < j < M Nk + S k For this range, M Nk − Sk + M Nk j + Mk > M Nk M Nk + S k , M Nk + S k M Nk + S k = M Nk+1 − M Nk − M Nk+1 − M Nk j + (M − 2)M k < M Nk+1 − 2M Nk + M Nk − S k , M Nk+1 − M Nk + S k − j M Nk+1 − Sk+1 and j + (M − 1)M k M Nk − S k + M M M Nk+1 + Nk+1 − Nk − S k + 2 M Nk+1 + Sk+1 M Nk+1 − M Nk + = ˆ k+1 (j+lM k ) = for l = 1, 2, , M −2, The first two sequences of inequalities give h from (3.8) The next two sequences imply that vk+1,j = vk+1,j+(M −1)M k = ck+1 M 2k+2 from (3.9) Hence Case (ii) of Proposition 2.11 holds Case M Nk + S k j M Nk − ˆ k+1 (j + lM k ) = for l = 0, 1, , M − This case is similar to Case with h 3.3 A General Family of Trigonometric Scaling Functions All the values of j are accounted for and so (2.15) holds for each k fore interpolatory wavelets ψkm , k 77 There- 1, m = 1, 2, , M − 1, of the form (2.4) exist To obtain the formula for the interpolatory wavelets, we follow the procedure described before Theorem 3.4 but using φˆk+1 (n) ein· φk+1 = n∈Z M ck+1 = M k+1 Nk+1 −Sk+1 ein· n=− M M Nk+1 +Sk+1 Nk+1 +Sk+1 −1 φˆk+1 (n) ein· + φˆk+1 (−n) e−in· + n= M Nk+1 −Sk+1 +1 instead of (3.4) For k = 0, φ1 , φ0 = c1 /M = T1m φ1 , φ0 and thus we have ψ0m = c0,m c1 (T1m φ1 − φ1 ) in place of (3.5) For k equals ck+1 M k+1 Tkν φk 2πµ M k+1 µ 1, the inner product Tk+1 φk+1 , Tkν φk and this leads to (2.27) in place of (3.6) Note that (2.27) also covers the case for k = Remark 3.7 First of all, the MRs found in Theorem 3.3 are special cases of the √ above corollary with ck = M k for every k Next, although the formula for the interpolatory wavelets happens to be the same as that obtained in the last part of Chapter 2, the intermediate calculations differ The formula should in fact be the same as we shall see in Proposition 3.11 that the two settings can coexist to give interpolatory wavelets From the formula, if every φk is a real-valued function, then every ψkm is also a real-valued function Besides MRs with interpolatory scaling functions and wavelets, there are those in our general family with other properties Corollary 3.8 Let c0 = 1, that is, φ0 is a constant function of value Suppose that for k 1, the following conditions hold 3.3 A General Family of Trigonometric Scaling Functions 78 (i) The Fourier coefficients φˆk (n), n ∈ Z, satisfy |φˆk (n)|2 = 1/M k when |n| M Nk − Sk , φˆk (n) = when |n| 1/M k when (ii) M M Nk+1 − Then {Vk }k Nk − S k < n < M M M Nk + Sk and |φˆk (n)|2 + |φˆk (n − M Nk )|2 = Nk + S k Nk − Sk+1 − Sk + is a multiresolution of L2 [0, 2π) with multiplicity and dilation M , having orthonormal {Tkl φk : l = 0, 1, , M k − 1} for every k Proof By Theorem 3.5, the sequence {Vk }k is a multiresolution of L2 [0, 2π) with multiplicity and dilation M For k = 0, {φ0 } is orthonormal trivially For k is orthonormal if and only if Mk (j) = p∈Z 1, {Tkl φk : l = 0, 1, , M k − 1} |φˆk (j + M Nk p)|2 = 1/M k for all j = 0, 1, , M k − 1, by Theorem 1.9 Indeed, we have the latter statement from the hypothesis The above corollary can also be viewed as a special case of the next result with A0 = B0 = which, as evident from its proof, gives A = B = Corollary 3.9 Let A0 and B0 be positive constants Suppose that for k 1, the following conditions hold (i) The Fourier coefficients φˆk (n), n ∈ Z, satisfy φˆk (n) = when |n| |φˆk (n)|2 Sk , A0 /M k |φˆk (n − M Nk )|2 (ii) M Nk+1 − Then {Vk }k B0 /M k when |n| B0 /M k when M M M Nk −Sk , and A0 /M k Nk − S k < n < Nk − Sk+1 − Sk + M M Nk + |φˆk (n)|2 + Nk + S k is a multiresolution of L2 [0, 2π) with multiplicity and dilation M , having the property that there exist positive constants A and B such that for every k 0, φk satisfies (2.11) for all cl ∈ C, l = 0, 1, , M k − Proof Condition (i) implies Condition (i) of Theorem 3.5, hence {Vk }k tiresolution of L2 [0, 2π) with multiplicity and dilation M is a mul- 3.3 A General Family of Trigonometric Scaling Functions Condition (i) also implies that A0 /M k 0, 1, , M k − In addition, we have v0,0 and B = max{B0 , c02 }, we obtain A/M k vk,j 2 B0 /M k , k 79 1, j = = c02 Letting A = min{A0 , c02 } vk,j B/M k , k 0, j = 0, 1, , M k − The rest of the proof follows from Proposition 1.23, noting that Mk (j) = vk,j for all k and j Remark 3.10 The above corollary covers the MRs in Theorem 3.3 with c0 = 1, A0 = A = 1/2 and B0 = B = All subsequent discussions are in the setting of Corollary 3.6 To obtain MRs with interpolatory scaling functions and wavelets for which {Tkl φk : l = 0, 1, , M k − 1} is orthonormal for each k 0, we combine the requirements on the Fourier coefficients from both Corollaries 3.6 and 3.8 This √ means c0 = and for k 1, ck2 /M 2k = 1/M k , that is, ck = M k Thus for k 1,    M  √ k if |n| Nk − S k , M φˆk (n) =   M 0 if |n| Nk + S k , and for M Nk − S k < n < M Nk + Sk , we must solve φˆk (n) + φˆk (n − M Nk ) = √ Mk (3.10) and |φˆk (n)|2 + |φˆk (n − M Nk )|2 = k M Using simple mathematics on complex numbers, these two equations can be shown to be equivalent to φˆk (n) and φˆk (n − M Nk ) lying on the circle in C with locus 1 (x − √ )2 + y = 4M k Mk √ and being symmetric about the centre 1/(2 M k ) (3.11) Knowing that interpolatory wavelets exist and the interpolatory scaling functions form orthonormal bases with their shifts, we analyze the possibility of the 3.3 A General Family of Trigonometric Scaling Functions 80 wavelets to satisfy similar orthonormality condition First observe that the orthogonal splines are, for k = the constant function v0,0 of value 1, and for k 1,   M  √ eij· if j Nk − S k ,  k  M   vk,j = φˆk (j)eij· + φˆk (j − M k )ei(j−M k )· if M Nk − Sk < j < M Nk + Sk , 2       √ ei(j−M k )· if M2 Nk + Sk j M Nk − Mk 2π·1 When M = 2, v0,0 ( 22π·1 0+1 ) and vk,j ( 2k+1 ), k 1, j = 0, 1, , 2k −1, can be verified √ to be nonzero Thus by choosing the interpolating heights ck,1 to be ck = 2k for every k 0, Theorem 2.12 assures that {Tkl ψk1 : l = 0, 1, , 2k − 1} is an orthonormal basis for Wk , k Consequently, {Tkl ψk1 : l = 0, 1, , 2k − 1} {φ0 } ∪ k is an orthonormal basis for L2 [0, 2π) However, when M > 2, it is impossible to satisfy Statement (ii) of Theorem 2.12 for any k The reason is that for k = or k with j ∈ {0, 1, , M Nk − Sk }, vk,j ( M2πµ k+1 ) is nonzero for more than one value of µ (in fact, all values of µ) taken from {1, 2, , M − 1} As a result, {Tkl ψkm : m = 1, 2, , M − 1, l = 0, 1, , M k − 1} is not orthonormal for all k ck,m , k 0, when M > Nevertheless, if 0, m = 1, 2, , M − 1, take appropriate values so that {ck,m } m M −1 M k+1 considered over all k max {ck,m } and m M −1 Mk are bounded below and above respectively by positive constants, Theorem 2.16 ensures that {Tkl ψkm : m = 1, 2, , M − 1, l = 0, 1, , M k − 1} {φ0 } ∪ k forms a Riesz basis for L2 [0, 2π) An example is ck,m = √ M k for all k and m This combination of Fourier coefficients requirements is summed up in the proposition below 3.3 A General Family of Trigonometric Scaling Functions Proposition 3.11 Let φ0 be a constant function of value Suppose that for k 1, the following conditions hold (i) The Fourier coefficient φˆk (n) equals M when |n| √1 Mk (ii) Nk + Sk ; and when M + 2√1M k eiθk,n and φˆk (n − M Nk ) = M Nk+1 − Then {Vk }k M √1 Mk when |n| M Nk − S k < n < √1 Mk Nk − Sk+1 − Sk + Nk −Sk and vanishes M Nk + Sk , φˆk (n) = − 2√1M k eiθk,n for some θk,n ∈ (−π, π] is a multiresolution of L2 [0, 2π) with multiplicity and dilation M , which has interpolatory scaling functions φk , k 0, satisfying (2.1) with interpo√ lating heights M k and {Tkl φk : l = 0, 1, , M k − 1} being orthonormal for each k, and interpolatory wavelets ψkm , k 0, m = 1, 2, , M − 1, of the form (2.4) √ and given by (2.27) In addition, if M = and ck,1 = 2k for every k 0, then {Tkl ψk1 : l = 0, 1, , 2k − 1} {φ0 } ∪ k is an orthonormal basis for L2 [0, 2π); if M > and ck,m , k 0, m = 1, 2, , M − 1, are selected so that {ck,m } max {ck,m } m M −1 M k+1 considered over all k and m M −1 Mk are bounded below and above respectively by positive constants, then {Tkl ψkm : m = 1, 2, , M − 1, l = 0, 1, , M k − 1} {φ0 } ∪ k forms a Riesz basis for L2 [0, 2π) Likewise, combining Corollary 3.6 and Corollary 3.9 gives a MR with interpolatory scaling functions and wavelets for which there exist positive constants A and B such that for every k 0, φk satisfies (2.11) for all cl ∈ C, l = 0, 1, , M k − This requires that for each k |n| M 1, the Fourier coefficient φk (n) vanishes when Nk + Sk , the inequalities A0 Mk ck Mk B0 , Mk 81 3.3 A General Family of Trigonometric Scaling Functions 82 or equivalently, A0 M k ck2 (3.12) Nk − Sk , and φˆk (n) and φˆk (n − M Nk ) M hold due to φk (n) in the range |n| B0 M k must satisfy ck φˆk (n) + φˆk (n − M Nk ) = k M when M M Nk −Sk < n < and A0 Mk |φˆk (n)|2 + |φˆk (n − M Nk )|2 B0 Mk Nk +Sk Using elementary mathematics on complex numbers like in the previous combination, an equivalent condition in the range M points about ck 2M k Nk + Sk is that φˆk (n) and φˆk (n − M Nk ) are symmetric M Nk − S k < n < in C and lie in the region x + iy ∈ C : 2A0 M k − ck2 4M 2k x− ck 2M k + y2 2B0 M k − ck2 4M 2k (3.13) Note from (3.12) that 2A0 M k − ck2 Thus for simplicity, we can take B0 2A0 so that 2A0 M k −ck2 4M 2k 0, m = 1, 2, , M − 1, are chosen such that Suppose that ck,m , k {ck,m } m M −1 ck+1 considered over all k (2A0 − B0 )M k max {ck,m } and m M −1 ck2 are bounded below and above respectively by positive constants Then all the hypotheses of Theorem 2.16 are justified and subsequently {Tkl ψkm : m = 1, 2, , M − 1, l = 0, 1, , M k − 1} {φ0 } ∪ k forms a Riesz basis for L2 [0, 2π) Letting A = min{A0 , c02 } and B = max{B0 , c02 } as in the proof of Corollary 3.9, it follows from (3.12) that AM k every k ck2 BM k for As such, choosing every ck,m to be ck gives {ck,m } m M −1 ck+1 AM k A = k+1 BM BM max {ck,m } and m M −1 ck2 BM k B = k AM A 3.3 A General Family of Trigonometric Scaling Functions 83 which fulfills the remaining hypothesis of Theorem 2.16 and results in a Riesz basis for L2 [0, 2π) As usual, there are also other options for ck,m , k 0, m = 1, 2, , M − For this combination, we have the following proposition Proposition 3.12 Let A0 and B0 be positive constants with B0 that for k 1, the following conditions hold (i) The Fourier coefficient φˆk (n) equals M ishes when |n| φˆk (n) = √ k ck 2M k 2A0 M −ck2 , 2M k (ii) M 2A0 Suppose when |n| M Nk + Sk ; and when + rk,n eiθk,n and φˆk (n − M Nk ) = √ 2B0 M k −ck2 and θk,n ∈ (−π, π] 2M k Nk+1 − Then {Vk }k ck Mk M Nk − Sk+1 − Sk + M Nk − Sk and van- Nk − S k < n < ck 2M k M Nk + S k , − rk,n eiθk,n for some rk,n ∈ is a multiresolution of L2 [0, 2π) with multiplicity and dilation M , having interpolatory scaling functions φk , k wavelets ψkm , k 0, of the form (2.1) and interpolatory 0, m = 1, 2, , M − 1, of the form (2.4) and given by (2.27) The scaling functions possess the property that there exist positive constants A and 0, φk satisfies (2.11) for all cl ∈ C, l = 0, 1, , M k − B such that for every k In addition, if ck,m , k 0, m = 1, 2, , M − 1, are selected so that {ck,m } m M −1 ck+1 considered over all k max {ck,m } and m M −1 ck2 are bounded below and above respectively by positive constants, then {Tkl ψkm : m = 1, 2, , M − 1, l = 0, 1, , M k − 1} {φ0 } ∪ k forms a Riesz basis for L2 [0, 2π) Remark 3.13 As noted in Remarks 3.7 and 3.10, the MRs constructed from the generalized de la Vall´ee Poussin means of the Dirichlet kernels in Theorem 3.3 are examples of the above proposition 3.3 A General Family of Trigonometric Scaling Functions 84 It is clear from the definition of the generalized de la Vall´ee Poussin means of the Dirichlet kernels ϕM N,S that these functions are real-valued and even The last focus of this thesis is to try to construct real-valued and even scaling functions of the types described in Propositions 3.11 and 3.12 We concentrate our analysis on the scaling functions from Proposition 3.11 The analysis of those scaling functions from Proposition 3.12 is similar Let us first deal with the scaling functions being real-valued Trivially, the constant function φ0 of value is real-valued For k 1, the Fourier coefficients given in the hypothesis of Proposition 3.11 clearly satisfy φˆk (n) = φˆk (−n) for n such that |n| M Sk and |n| M Nk − Sk < |n| < M Nk + Sk to satisfy φˆk (n) = φˆk (−n) in order for φk to be real-valued The average of Nk +Sk is M range M Nk When M is even, Nk − S k < n < and this translates into φˆk √1 Mk ± √i Mk M M Nk Nk + Sk Thus we need φˆk M Nk M Nk − Sk and is an integer right in the middle of the M Nk = φˆk − M2 Nk and φˆk − M2 Nk taking distinct values from which are the highest and lowest points of the circle given by (3.11) When M is odd, there is no such middle integer between M Nk − Nk + Sk According to Proposition 1.3(a), we also need the Fourier coefficients for n in the range M M M Nk − Sk and Nk +Sk For both even and odd M , the remaining Fourier coefficients are such that for M Nk −Sk < n < M Nk , φˆk (n), φˆk (n−M Nk ), φˆk (−n) and φˆk (−n+M Nk ) lie on the circle (3.11) and form vertices of a rectangle which is centred at √1 , Mk has sides parallel to the vertical and horizontal axes, and has {φˆk (n), φˆk (n−M Nk )} and {φˆk (−n), φˆk (−n+M Nk )} as pairs of opposite vertices Note that the rectangle may be degenerate in the sense that the corners are only two points which are either √1 Mk ± √i Mk or 0, √M k (the leftmost and rightmost points of the circle) We now discuss whether even scaling functions can be constructed Obviously, φ0 is even and for k M Nk − Sk and |n| 1, φˆk (n) = φˆk (−n) is satisfied for n such that |n| M Nk + Sk Due to Proposition 1.3(b), the remaining requirement for φk to be even is that φˆk (n) = φˆk (−n) when M Nk − Sk < |n| < 3.3 A General Family of Trigonometric Scaling Functions M Nk + Sk Similar to above, M Nk +Sk when M is even In this case, φˆk ( M2 Nk ) = φˆk (− M2 Nk ) and (3.10) imply that φˆk ( M2 Nk ) = φˆk (− M2 Nk ) = M Nk 85 √1 , Mk is an integer right between M Nk − Sk and a contradiction to the fact that the Fourier coefficients lie on the circle with locus (3.11) Therefore the scaling functions described in Proposition 3.11 cannot be even when M is even However, no such contradiction exists for the case when M is odd, and for M Nk − S k < n < M Nk , φˆk (n) and φˆk (−n) are the same point on the circle (3.11) with φˆk (n − M Nk ) and φˆk (−n + M Nk ) being the point on the opposite end of the circle When M is odd, merging the requirements for real-valued and even scaling functions of the type in Proposition 3.11 entails that, in addition to the other common √ conditions, each scaling function φk , k 1, must satisfy φˆk (n) = φˆk (−n) = 1/ M k and φˆk (n−M Nk ) = φˆk (−n+M Nk ) = 0, or vice versa, for M Nk −Sk < n < M Nk The next two examples illustrate the above discussion Example 3.14 For even M and k 1, let    √1   Mk       √1 k + √i k M φˆk (n) = M   √1  − 2√iM k  k M      0 if |n| < if n = M Nk M Nk , , if n = − M2 Nk , if |n| > M Nk Then φk is a real-valued but not even scaling function, satisfying the hypothesis of Proposition 3.11 Indeed, φˆk (n) ein· φk = n∈Z =√ Mk Nk −1 ein· + n=−( M Nk −1) i √ − √ Mk Mk  M + =√ M 1 Mk Nk −1 1 + n=1 i √ + √ Mk Mk M ei Nk · M e−i Nk ·  cos(n·) + √ cos Mk M Nk · − √ sin Mk M Nk · , 3.3 A General Family of Trigonometric Scaling Functions 86 where we see from the last expression that φk is real-valued but not even Example 3.15 For odd M and k 1, let     √ k if |n| M ˆ φk (n) =   0 if |n| M Nk − 12 , M Nk + 12 Then φk is a real-valued and even scaling function satisfying the hypothesis of Proposition 3.11 Indeed, φk = √ M Mk Nk − 12 ein· n=−( M Nk − 12 )  = √ Mk M  Nk − 12 1 + cos(n·) n=1 which is clearly real-valued and even The additional conditions for the scaling functions in Proposition 3.12 to be realvalued is as follows, where k When M is even, φˆk are points in (3.13), joined vertically with mid-point M , and M Nk − S k < n < M Nk , ck 2M k M Nk and φˆk − M2 Nk For both even and odd the points φˆk (n), φˆk (n − M Nk ), φˆk (−n) and φˆk (−n+M Nk ) lie in (3.13) and form vertices of a rectangle which is centred at ck , 2M k has sides parallel to the vertical and horizontal axes, and has {φˆk (n), φˆk (n−M Nk )} and {φˆk (−n), φˆk (−n + M Nk )} as pairs of opposite vertices The rectangle may degenerate into a vertical or horizontal line For the scaling functions in Proposition 3.12 to be even, when M is even, the annulus (3.13) must be the circular region x + iy ∈ C : x − because the points φˆk region, M ck 2M k M Nk ck 2M k + y2 2B0 M k − ck2 4M 2k and φˆk − M2 Nk are exactly the centre of the circular The annulus need not be a circular region for odd M For all M and Nk − S k < n < M Nk , φˆk (n) and φˆk (−n) are the same point in (3.13) with φˆk (n − M Nk ) and φˆk (−n + M Nk ) being the opposite point symmetric about ck 2M k 3.3 A General Family of Trigonometric Scaling Functions 87 Combining the above two cases, to have real-valued and even scaling functions of the kind in Proposition 3.12, for M Nk −Sk < n < M Nk , φˆk (n) and φˆk (−n) are the same point on the real-axis in (3.13) while φˆk (n − M Nk ) and φˆk (−n + M Nk ) are the point on the real-axis whose mid-point with φˆk (n) or φˆk (−n) is ck 2M k We conclude the thesis by summarizing the various combinations of properties found for our family of trigonometric scaling functions Proposition 3.11 gives us a method to find, from this family, scaling functions which are interpolatory and, at the same time, their shifts form orthonormal bases for the multiresolution subspaces In addition, interpolatory wavelets exist and they can be chosen, based on their interpolating heights, to form orthonormal bases for L2 [0, 2π) if M = 2, and, at best, Riesz bases for L2 [0, 2π) if M > For all M 2, these scaling functions can also be real-valued but they can only be even when M is odd Proposition 3.12 gives interpolatory scaling functions from this family which are more general than those from Proposition 3.11 in the sense that the shifts of the scaling functions need not form orthonormal bases However, interpolatory wavelets still exist and their interpolating heights can be properly chosen so that the wavelets form Riesz bases for L2 [0, 2π) For all M 2, real-valued scaling functions of this type exist Even scaling functions of 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biorthogonal multiwavelet systems with compact support, Appl Comput Harm Anal 11 (2001), 420–438 89 [...]... can verify the 1.3 Periodic Wavelets from Polyphase Splines 12 proof for themselves by replacing every incidence of 2k by M k in Lemmas 3.9, 3.10, 3.11 and Theorem 3.12 of [7] as well as adjusting the proof of Theorem 3.12 of [7] slightly to cater to multiple scaling functions Theorem 1.14 Let Vk , k 0, be subspaces of L2 [0, 2π) satisfying MR1 and MR2, each subspace with scaling functions φm k , m... those of Proposition 1.5 and Corollary 1.13 Lemma 1.16 Let H and G be p×n and q×n matrices respectively, where p+q = n Suppose that both matrices are of full rank and HG∗ = 0p×q Then H G is invertible Proof Let {vi : i = 1, 2, , p} and {uj : j = 1, 2, , q} represent the rows of H and G respectively Then both sets are linearly independent and vi , uj = 0 for each pair of i and j Consider the equation... (0) is a positive definite matrix, its eigenvalues are positive If λ1 and λr are the largest and smallest eigenvalues respectively, then by the previous lemma, for any x ∈ Cr , A xx∗ x M0 (0) x∗ B xx∗ , 1.3 Periodic Wavelets from Polyphase Splines 21 where A := min{A, λr } and B := max{B, λ1 } Note that A and B are positive Fix k 0 and j ∈ {0, 1, , M k − 1} Define u˜m k,j , m = 1, 2, , r(M − 1),... , M k − 1 l=0 1.3 Periodic Wavelets from Polyphase Splines 28 (ii) There exist positive constants A and B such that for every k 0, j = 0, 1, , M k − 1, and y ∈ Cr(M −1) , A yy ∗ Mk M −1 l=0 and y Nk (j) y ∗ B yy ∗ Mk ∗ Gk+1 (j + lM k )Mk+1 (j + lM k )Hk+1 (j + lM k ) = 0r(M −1)×r (iii) There exist positive constants C1 and C2 such that for every k 0, j = 0, 1, , M k − 1, and y ∈ Cr(M −1) ,... (j)Yk (j)∗ y ∗ k+1 M = and similarly A y Yk (j)Yk (j)∗ y ∗ M k+1 y Nk (j) y ∗ Under (ii), A yy ∗ k M and so A B B y Yk (j)Yk (j)∗ y ∗ k+1 M y Nk (j) y ∗ y Yk (j)Yk (j)∗ y ∗ Likewise y Yk (j)Yk (j)∗ y ∗ M yy ∗ B A M yy ∗ and (iii) holds with C1 = A /B and C2 = B /A Conversely, (ii) follows from (iii) with A = AC1 and B = BC2 Fix k 0 and j ∈ {0, 1, , M k − 1} In terms of Xk (j) and Yk (j), (ii) implies... the criteria for the existence of this type of wavelets Theorem 1.25 Suppose that there exist positive constants A and B such that for 0, the scaling functions of Vk , φm k , m = 1, 2, , r, satisfy Statement (i) every k of Proposition 1.23 The following statements are equivalent (i) There exist positive constants A and B such that for every k 0, the wavelets of Wk , ψkm , m = 1, 2, , r(M − 1),... Xk (j)Dk (j) and Yk (j) := Yk (j)Dk (j) Note that Mk+1 (j + lM k ), j = 0, 1, , M k − 1, l = 0, 1, , M − 1, are positive 1 definite This ensures that all Mk+1 (j + lM k ) 2 and thus all Dk (j), Xk (j) and Yk (j) ∗ ∗ are well defined It is obvious that Mk (j) = Xk (j)Xk (j) and Nk (j) = Yk (j)Yk (j) from Corollary 1.13 and (1.11) for each j Here are the criteria for the existence of wavelets Theorem... −1 l=0 and y Yk (j)Yk (j)∗ y ∗ C2 M yy ∗ ∗ Gk+1 (j + lM k )Mk+1 (j + lM k )Hk+1 (j + lM k ) = 0r(M −1)×r (iv) There exist positive constants C1 and C2 such that for every k 0, j = 0, 1, , M k − 1, and z ∈ CrM , C1 M zz ∗ M −1 l=0 and z Xk (j) Yk (j) Xk (j) Yk (j) ∗ z∗ C2 M zz ∗ ∗ Gk+1 (j + lM k )Mk+1 (j + lM k )Hk+1 (j + lM k ) = 0r(M −1)×r Proof From (1.3) and (1.4), all Tkl ψkm ∈ Wk if and only... together with Proposition 1.23, proves the equivalence of (i) and (ii) Fix k 0 and j ∈ {0, 1, , M k − 1} The hypothesis on the scaling functions amounts to A xx∗ k M ∗ x Mk (j) x∗ = x Xk (j)Xk (j) x∗ B xx∗ k M for all x ∈ Cr For y ∈ Cr(M −1) , M −1 y Nk (j) y ∗ ∗ y Gk+1 (j + lM k )Mk+1 (j + lM k )Gk+1 (j + lM k ) y ∗ = l=0 1.3 Periodic Wavelets from Polyphase Splines B M k+1 29 M −1 ∗ y Gk+1 (j +... will be used again in the next chapter when we discuss interpolatory scaling functions For fixed k 0 and j ∈ {0, 1, , M k − 1}, we make the notation Mk (j) := m m vk,j , vk,j r (1.5) m,m =1 m As Mk (j) is the Gram matrix of {vk,j : m = 1, 2, , r}, it is positive definite if and m only if {vk,j : m = 1, 2, , r} is linearly independent, and positive semi-definite in any case The reader is referred ... scaling functions and wavelets in L2 (R) in the literature include [1], [5], [8], [14] and [16] Interpolatory periodic scaling functions and wavelets have been discussed in [2], [11], [10] and [12],... study of periodic multiresolutions with general dilation, together with their corresponding scaling functions and wavelets From this analysis of general periodic scaling functions and wavelets, ... independently and shown to form interpolatory periodic scaling functions in Section 3.2 In Section 3.3, a sharp observation of these modified functions leads to an extensive family of periodic scaling functions,