Spectral properties of transfer operators

SPECTRAL PROPERTIES OF TRANSFER OPERATORS Gao Xiaojie (M.Sc., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements I am grateful to a number of people for their contributions to my thesis work and life in Singapore over the past few years. I would like to thank Professor Lee Seng Luan, my supervisor, whose work was the inspiration for this project, for his many suggestions and constant support during this research. Here, I particularly thank him for trusting me to plan my own course and for giving me the freedom and encouragement to pursue my research goals. His patience with my errors and encouragement of my ambitions was far more than I expected. It was my good fortune to have worked with such a diverse mathematician who is also an expert in my technical field. I am also indebted to Dr. Sun Qiyu whose invaluable support has played a key role to go through these challenging years. I attended a lot of seminars regularly conducted by the Department of Mathematics and the Centre for Wavelets, Approximation and Information Processing (CWAIP), which enriched me with valuable insights and inspirations. I want to thank the regular speakers: Goh Say Song, Timothy Nicholas Trevin Goodman, Jia Rong-Qing, Wayne M. Lawton, Charles A Micchelli, Shen Zuowei, Tang Wai Shing, Zhou Ding Xuan. I enjoyed the discussions with visitors to the Department of Mathematics and CWAIP: Antoine Ayache, Victor D Didenko, Yang JianSheng, Ye Yuan Ling. I attended a number of courses, and would like to thank Jesudason J. Packer and Lou Jiann Hua for making them both enjoyable and informative. I have enjoyed friendly and stimulating surroundings in CWAIP. I would like ii to thank many current and former colleagues: Ding Ping, Jiang Qingtang, Li Hui, Liang Yong Qing, Lim Zhi Yuan, Lin Jianyu, Lin Zhenyong, Liu Bao, Liu Chao Qiang, Shang Fuchun, Shen Lixin, Tan Hwee Huat, Tham Jo Yew, Wai Kok Hoong, Wang Pei Jie, Wang Yuping, Xia Tao, Zhang Zhuosheng. Of course, I am grateful to my parents, my grandmother, my wife and my sister for their patience and love. Without them this work would never have come into existence. Finally, I wish to thank the following: Chen Feng, Dai Bo, Li Zhao, Tan Boon Huat, Tang Hongyan, for their friendship. Singapore September 17, 2003 Gao Xiaojie iii Table of Contents Table of Contents iv Summary vi Introduction 0.1 Multiscale operators . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Transition operators and subdivision operators . . . . . . . . . . . . 0.3 Scaling operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Spectrum of Multiscale Operators on Weighted Lp Spaces 1.1 Boundedness of Wc,α on Lpw (R) . . . . . . . . . . . . . . . . . . . . 1.2 Spectrum of Wc,α on Lpw (R) . . . . . . . . . . . . . . . . . . . . . . 1.3 Characterization of compactness of Wc,α on Lpw (R) . . . . . . . . . . 1.4 Eigenfunctions of a continuous refinement operator and its adjoint . 1.4.1 Eigenfunctions of a continuous refinement operator and its adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Construction of the polynomial eigenfunctions . . . . . . . . 1.5 Two sequences of multiresolution like subspaces . . . . . . . . . . . 12 15 27 31 Transition Operators on Weighted 2.1 Boundedness . . . . . . . . . . . 2.2 Compactness . . . . . . . . . . . 2.3 Trace property . . . . . . . . . . Sequence Spaces 41 . . . . . . . . . . . . . . . . . . . 42 . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . . . . . . 48 Spectrum of Subdivision Operators 3.1 Boundedness . . . . . . . . . . . . 3.2 Resolvent sets . . . . . . . . . . . . 3.3 Residual spectrum . . . . . . . . . 3.4 Proof for Theorem 3.0.5 . . . . . . 3.5 Spectrum of transition operators . . iv 31 34 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 55 57 61 68 74 Eigenvalues and Biorthogonal Eigensystems of Scaling Operators 82 4.1 Scaling operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Eigenvalues of scaling and transition operators . . . . . . . . . . . . 90 4.3 Scaling operators and B-splines . . . . . . . . . . . . . . . . . . . . 93 4.3.1 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . 95 4.3.2 Proof of Theorem 4.3.2 . . . . . . . . . . . . . . . . . . . . . 96 4.4 Appell sequences and adjoint scaling operators . . . . . . . . . . . . 98 4.5 Biorthogonal eigensystems involving Bernoulli and Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Bibliography 106 v Summary Multiscale, transition, subdivision and scaling operators arise from various problems ranging from wavelets, multiscale analysis, approximation and dynamical systems to geometric modelling. They share a common general form Tµ,α f (x) = α f (αx − t)dµ(t), R where µ is a finite Borel measure and α > 1. The application of these operators, which we call as transfer operators in general, has gained a lot of ground in recent years, especially due to the development and growth of information science. It is known that the spectral properties of each operator have close ties with its application in the related field and in many cases, the estimation of performance is characterized by the spectrum. As a result, a lot of work have been done on the spectral study of the operators. However, due to the wide range of applications, problems are considered under different settings or for different purposes. Thus the existing results are inconsistent and are not comparable due to the gaps present among them. This brings about the aim of the current work: To unify the known results in a general setting. Our purpose in this thesis is to investigate the spectral properties of these operators. We study the relationship of the operators and give an explicit description of the spectrum of each operator in different spaces. After discussing the boundedness of each operator, we begin by establishing the spectral description of the operator in weighted spaces, which includes many vi vii of the known results as special cases. Compared with the available knowledge in the literature, these descriptions are shown to be more precise and complete. It is worth pointing out that, by studying the operators in weighted spaces, we bridge the gaps present among the existing results on the spectral description of the operators. We also study other spectral properties of the operators. Among these, compactness and trace properties of the operators are characterized with the known results being listed as special cases. A set of conjugate eigensystem of adjoint operators is set up as a byproduct of the spectral analysis. Relationships between the spectra of different operators are also established and those between transition operators and scaling operators are characterized in terms of B-splines. Besides focusing on the spectral properties, we also discuss other related aspects of these operators. As an example, the convergence of a sequence of Appell polynomials is illustrated with the help of the eigensystem of the scaling operator. Introduction Transfer operators arise from various problems ranging from wavelets, multiscale analysis and dynamical systems to geometric modeling. The objective of this thesis is to study the spectral properties of these operators and discuss related results. 0.1 Multiscale operators The linear multiscale representation c(α(t − x))f (t)dt, α = 0, f ∈ Lq , M f (α, x) = α R provides an efficient tool for scale-space analysis of information. If c satisfies an admissibility condition then M is a continuous wavelet transform. Popular classical choices for c are the Gaussian functions and their derivatives as well as the modulated Gaussian. Recent popular choices for c are the uniform B-splines and their derivatives. For a fixed scale α and integrable function c let ∗ f (x) := M f (α, x/α) Wc,α c(αt − x)f (t)dt, f ∈ Lq . = α R Its adjoint is Wc,α f (x) = α c(αx − t)f (t)dt R f (αx − t)c(t)dt, f ∈ Lp . = α R The operator Wc,α , which will be called a multiscale operator, arises in many different unrelated contexts. The simplest operator of this kind is one with kernel c = 21 χ(−1,1] and dyadic scale α = 2, and it was studied independently by Kabaya and Iri [44] in conjunction with the error analysis in the computation of highly complicated functions, and by Rvachev [55] for the approximation of functions. Recent interests in multiscale operators are associated with nonstationary multiresolution and wavelets [10, 27, 29], nonstationary subdivision processes [14, 17], and invariant densities for model sets in the study of quasicrystals [1, 2]. 0.2 Let Transition operators and subdivision operators be the space of all sequences on Z and the space of all finitely supported ones. Endowed with the usual topologies they are linear topological spaces. For a fixed real-valued sequence a := {a(k)}k∈Z with finite support, we define the transition operator Ta and subdivision operator Sa on Ta v(i) := by a(2i − j)v(j), (0.2.1) a(i − 2j)v(j), (0.2.2) j∈Z and Sa v(i) := j∈Z where v := {v(j)}j∈Z ∈ . The transition operator Ta and the subdivision operator Sa are both continuous on as well as on 0, and the subdivision operator Sa corresponding to a sequence a is the conjugate of the transition operator Ta corresponding to the sequence a ˜ := a(−k) k∈Z in the sense that Ta v, w = v, Sa˜ w for any pair (v, w) ∈ v, w := × or × 0, where v(k)w(k) for v := {v(k)}k∈Z and w := {w(k)}k∈Z . k∈Z The transition operator is closely related to Ruelle operator R on C(X) in dynamical systems [21, 34, 53, 54], where N Rf (x) = pi (Si (x))f (Si (x)), f ∈ C(X), i=1 Si are contractive maps and pi , ≤ i ≤ N , are continuous weight functions on a compact metric space (X, d). Here C(X) is the space of all continuous functions on X, and a contractive map S on X is one that satisfies d(S(x), S(y)) ≤ ρd(x, y) for all x, y ∈ X, for some < ρ < 1. For a sequence v := {v(k)}k∈Z , the formal sum k∈Z v(k)z k is its Z-transform to be denoted by Z(v)(z). In the Fourier domain the operators corresponding to the transition operator Ta and of the subdivision operator Sa , to be denoted by Ta and Sa respectively, are given by Ta f (ξ) := Z(Ta v)(e−iξ ) = H(ξ/2)f (ξ/2) + H(ξ/2 + π)f (ξ/2 + π) (0.2.3) and Sa f (ξ) := Z(Sa v)(e−iξ ) = H(ξ)f (2ξ), (0.2.4) where H(ξ) = 12 Z(a)(e−iξ ) and f (ξ) := Z(v)(e−iξ ). Expressed in the form Ta f (ξ) = H(S1 (ξ))f (S1 (ξ)) + H(S2 (ξ))f (S2 (ξ)), with contractive maps S1 and S2 defined on [0, 2π] by Si (ξ) = ξ/2 + (i − 1)π, i = 1, 2, (0.2.5) 97 and σe (Ta , L2c ) = σe (Tτl a , L2c ). So we can assume without loss of generality that Ia = {0, . . . , N } for some positive integer N . That (a) implies (b) is the result of Theorem 4.3.1, while the equivalence between (c) and (b) follows from (4.2.1) and (4.3.4) and the fact that ∈ σe (Ta , L2c ). Now we show that (b) implies (a). By Lemma 4.2.3, σe (Ta , L2c ) = ∅ or {1, 1/2, . . . , 1/2k0 } for some non-negative integer k0 . Since a(0), a(N ) ∈ σe (Ta , (Ia )) by the matrix representation of Ta , it follows that σe (Ta , (Ia )) = {1, 1/2, . . . , 1/2k0 }. (4.3.8) Since as an operator on (Ia ), Ta is represented by the matrix (a(2i − j))N i,j=0 , its eigenpolynomial is of the form (−1)N +1 λN +1 + (−1)N trace(Ta )λN + g(λ), where trace(Ta ) = N j=0 a(j) = and g is a polynomial of degree less than N. Thus the sum of all the eigenvalues of Ta , counting multiplicity is equal to trace(Ta ) = 2. Therefore, assuming that 1/2k is an eigenvalue of Ta on (Ia ) with multiplicity k0 k k=0 lk /2 lk ≥ 1, we obtain the equations = and k0 k=0 lk = N + 1. The first equation yields lk = for k = 0, 1, . . . , k0 − 1, lk0 = 2. The second equation then implies k0 = N − 1. (4.3.9) By (4.3.8), (4.3.9) and the assumption (b), we obtain σe (Ta , L2c ) = {1, 1/2, · · · , 1/2N −1 }. This together with Lemmas 4.2.3 and 4.2.4 and the fact Ia = {0, . . . , N } prove that H(ξ) := −ikξ = k∈Z a(k)e 1+e−iξ N , and hence the assertion (a). ✷ 98 4.4 Appell sequences and adjoint scaling operators Let a := {a(k)}k∈Z ∈ satisfy k∈Z a(k) = 2, and let φ be the unique compactly supported distributional solution of the refinement equation Ta φ = φ with φ(0) = 1. Taking distributional derivatives on both sides of the refinement equation Ta φ = φ shows that for any positive integer n, φ(n) is an eigenfunction of the scaling operator Ta on (C ∞ ) associated with the eigenvalue 2−n , n ∈ Z+ . Since φ is compactly supported, its Fourier transform φ is analytic. Therefore, it generates ∞ a pair of Appell sequences {Pn }∞ n=0 and {Qn }n=0 via the generating functions ∞ φ(iz)e−xz = zn n! (4.4.1) zn . n! (4.4.2) Pn (x) n=0 and exz φ(iz) ∞ = Qn (x) n=0 Theorem 4.4.1. Let {Pn } and {Qn } be Appell sequences generated by φ in (4.4.1) and (4.4.2) respectively. Then for n ∈ Z+ , Ta Pn = 2n+1 Pn , (4.4.3) and ∗ Ta Qn = 2−n Qn . (4.4.4) 99 Proof. We shall prove only (4.4.3), since the proof of (4.4.4) is similar. ∞ Ta Pn (x) n=0 zn = n! ∞ a(k)Pn (2x − k) n=0 k∈Z zn n! ∞ = a(k) Pn (2x − k) n=0 k∈Z zn n! a(k)φ(iz)e−(2x−k)z = k∈Z = 2φ(iz)H(iz)e−2xz = 2φ(2iz)e−x(2z) ∞ 2n Pn (x) = n=0 zn , n! which leads to (4.4.3). ∗ The relation (4.4.4) shows that Qn is an eigenfunction of Ta with eigenvalue 2−n , n ∈ Z+ . Recall that φ(n) is an eigenfunction of Ta with the same eigenvalue 2−n for each n. These observations motivate the following ∞ Theorem 4.4.2. The eigensequences {(−1)n φ(n) }∞ n=0 and {Qn /n!}n=0 form a biorthog- onal system, i.e. (−1)m φ(m) , Qn /n! = δm,n , m, n ∈ Z+ . (4.4.5) where ·, · denotes the action of a distribution on a test function. Proof. The relation (4.4.5) follows from ∞ m (m) (−1) φ n=0 zn (−1)m φ(m) , e , Qn = n! φ(iz) ·z = zm. ∗ We end this section with results on the eigenvalues of Ta on the spaces S, C ∞ and S . 100 Theorem 4.4.3. Let a := {a(k)}k∈Z ∈ k∈Z satisfy k∈Z a(k) = 2, and set H(ξ) := a(j)e−ikξ . Then ∗ (i) σe (Ta , S) = ∅. ∗ (ii) ∈ σe (Ta , C ∞ ) if and only if H is not a monomial. ∗ (iii) σe (Ta , C ∞ ) \ {0} = {2−n : n ∈ Z+ }. ∗ (iv) σe (Ta , S ) = C \{0} if H(ξ) = for all ξ ∈ R. ∗ Proof. We first prove that σe (Ta , S) = ∅. The idea is similar to that in the proof of assertion (4.1.3). Take any complex number λ and let f be a Schwartz function that satisfies ∗ Ta f = λf. (4.4.6) Taking the Fourier transform on both sides of the equation (4.4.6) gives 2H(−ξ)f (2ξ) = λf (ξ). (4.4.7) Thus f (ξ) = λ(2H(−ξ/2))−1 f (ξ/2) on a neighborhood of the origin, say {ξ : |ξ| ≤ δ}. Then using the same argument as in the proof of the assertion (4.1.3), we conclude that either f (ξ) ≡ on that neighborhood, or f (ξ) = Pk (ξ)/φ(−ξ) on some neighborhood of the origin, where Pk (ξ) is a nonzero monomial of degree k, and φ is the refinable function that satisfies Ta φ = φ with φ(0) = 1. In the first case, f (ξ) ≡ for all ξ ∈ R by (4.4.7), and hence f ≡ 0. In the second case, λ = 2k+1 and f (ξ) = Pk (ξ)/φ(ξ) for all ξ ∈ R by an iterative application of (4.4.7). Thus = Pk (ξ) = f (ξ)φ(ξ), which is a contradiction since the right hand side f (ξ)φ(ξ) tends zero as ξ tends to infinity. The decay of f (ξ)φ(ξ) at infinity follows from the facts that f is a Schwartz function and that φ is dominated by a ∗ polynomial. This completes the proof of the assertion σe (Ta , S) = ∅. 101 ∗ To prove (ii), it suffices to prove that ∈ σe (Ta , C ∞ ) if H is a monomial, ∗ and that ∈ σe (Ta , C ∞ ) if H is not a monomial. The first assertion follows easily from the definition of the adjoint scaling operator. To prove the second assertion, we assume that H is not a monomial. By the Fundamental Theorem of Algebra it has a nonzero root ξ0 ∈ C. Set f (x) = ei2ξ0 x . Clearly f ∈ C ∞ and ∗ Ta f (x) = ∗ k∈Z a(k)eiξ0 k eiξ0 x = H(ξ0 )eiξ0 x = 0, which leads to ∈ σe (Ta , C ∞ ). This completes the proof of assertion (ii). To prove (iii), we recall from Theorem 4.4.1 that Qn ∈ C ∞ is an eigenfunction ∗ of Ta with eigenvalue 2−n , n ∈ Z+ . Therefore, we need only to show that σe (Ta , C ∞ )\{0} ⊂ {2−n : n ∈ Z+ }. ∗ (4.4.8) ∗ To this end, let λ ∈ σe (Ta , C ∞ )\{0} and fλ ∈ C ∞ satisfy Ta fλ = λfλ . Let φ be the unique compactly supported distributional solution of the refinement equation φ= k∈Z a(−k)φ(2 · −k) with φ(0) = 1, and denote the convolution between fλ and φ by Fλ , i.e. Fλ (x) = fλ (x − ·), φ . Then Fλ ∈ C ∞ and a(−k) fλ (x − ·), φ(2 · −k) Fλ (x) = k∈Z a(−k) fλ (x − k/2 − ·), φ(2·) = k∈Z ∗ = Ta fλ (2x − 2·), φ(2·) = λ fλ (2x − ·), φ = λFλ (2x). Let g be the restriction of the function Fλ on the interval [1, 2]. It then follows from Fλ (x) = λg(2x) that Fλ (x) = λn g(2n x) for all x ∈ [2−n , 2−n+1 ]. Recall that Fλ (x) ∈ C ∞ , and that the k-th derivative of Fλ on 2−n [1, 2] is (2k λ)n g (k) (2n x). Therefore, for any integer k such that 2k |λ| > 1, we must have g (k) (x) = because (k) of the continuity of Fλ at the origin. Thus g(x) is a polynomial. Note that the right k-th derivative of Fλ at the point is g (k) (1), and that the left k-th derivative 102 of Fλ at is λ2k g (k) (2) for any nonnegative integer k. Thus g (k) (1) = λ2k g (k) (2) for all k ≥ 0. Writing g(x) = L j=0 (4.4.9) aj xj , where aL = 0, and using (4.4.9) with k = L, we obtain λ = 2−L . This establishes the inclusion (4.4.8), and hence (iii). Finally we prove (iv) using a similar argument as that in Theorem 4.1.1. Let φ satisfy Ta φ = φ. Then φ(ξ) = ∞ j=1 H(2−j ξ), which together with our assumption that H is non-vanishing lead to |φ(ξ)| ≥ C(1 + |ξ|)−N for all ξ ∈ R, (4.4.10) where C and N are positive constants. Take a nonzero complex number λ and hλ (ξ) , φ(−ξ) λ k k∈Z define fλ (ξ) = h, hλ = where hλ is a nonzero tempered distribution defined by h, g(2−k ξ) for any h ∈ S, and g is a bounded measur- able function supported in {ξ : < |ξ| < 2} and satisfies R ξ l g(ξ)dx = for all ≤ l ≤ − ln2 |λ|. We remark that hλ is a measurable function if |λ| > 1. By the same argument as in the proof of Theorem 4.1.1, hλ is a nonzero tempered distribution, which together with the estimate (4.4.10) imply that fλ is a nonzero tempered distribution. For any h ∈ S, we have h, Ta fλ = h, 2H(− ·)(φ(−2 ·))−1 hλ (2 ·) = k∈Z = λ k∈Z λ k λ k −1 ˆ h(φ(−·)) , g(2−k+1 ·) −1 ˆ h(φ(−·)) , g(2−k ·) = λ h, fλ , ∗ and hence Sa fλ = λfλ . This proves that C \{0} ⊂ σe (Ta , S ). Therefore it remains ∗ ∗ to prove ∈ σe (Ta , S ). Suppose, on the contrary, that ∈ σe (Ta , S ). Then ∗ H(−ξ)f0 (2ξ) = Ta f0 (ξ) = (4.4.11) 103 for some nonzero tempered distribution f0 . Multiplying both sides of the equation (4.4.11) by the bounded C ∞ function H(−ξ)−1 gives f0 = 0, which is a contradiction. 4.5 Biorthogonal eigensystems involving Bernoulli and Hermite polynomials For the uniform B-spline BN of order N , we have BN (ξ) = {N } ∞ }n=0 generating function (4.4.2) for the Appell sequence {Bn ∞ Bn{N } (x) n=0 zn = n! {N } which shows that the polynomials Bn z z e −1 1−e−iξ iξ N . Thus the becomes N ezx , (4.5.1) , n ∈ Z+ , are indeed Bernoulli polynomials of order N . From Theorem 4.4.2 it follows that for each N ∈ N the Bernoulli {N } polynomials Bn (n) , n ∈ Z+ , are biorthogonal to the distributional derivatives BN , n ∈ Z+ , of the N -th order B-spline, a result that appears to be a new link between the two well-known functions. ˜N and the normalized Bernoulli polynoWe define the normalized B-splines B ˜n{N } by mials B ˜N (x) := σN BN σN x + N B (4.5.2) ˜ {N } (x) := σ −n B {N } σN x + N B n n N (4.5.3) and respectively, where σN = N . 12 Clearly, the biorthogonal relationship is preserved under the normalization, i.e. ˜ ,B ˜ {N } /n! = δm,n , m, n ∈ Z+ . (−1)m B n N (m) (4.5.4) 104 For the Gaussian G(x) := √1 e−x /2 , 2π its Fourier transform G(ξ) = e−ξ /2 is still a Gaussian, and it generates the Hermite polynomials in the same way as the B-splines generate the Bernoulli polynomials in (4.5.1), i.e. exz G(iz) ∞ = Hn (x) n=0 zn . n! (4.5.5) Since G(n) = (−1)n Hn G, n ∈ Z+ , the orthonormality of the Hermite polynomials {Hn } with respect to the weight G is equivalent the biorthogonality of the sequences {(−1)n G(n) } and {Hn /n!}, (−1)m G(m) , Hn /n! = δm,n , m, n ∈ Z+ . (4.5.6) The uniform B-spline BN is the probability density function of the sum of N copies of independent identically distributed uniform random variables on the interval [0, 1), and hence ˜N (x) = G(x), x ∈ R, lim B N →∞ (4.5.7) by Central Limit Theorem (see [13], [8], [62]). While the Gaussian and its derivatives have many applications, ranging from Brownian motion to scale-space representation ([49, 64]), the B-splines provide fast computational algorithms. Equations (4.5.4), (4.5.6) and (4.5.7) suggest that for each n the normalized ˜n{N } converge to the Hermite polynomials Hn as N → ∞. Bernoulli polynomials B In particular, we have ˜n Theorem 4.5.1. Let B {N } be the normalized Bernoulli polynomial of degree n and order N , and Hn be the Hermite polynomial of degree n. Then, for every n ∈ Z+ and x ∈ R, ˜ {N } (x) = Hn (x). lim B n N →∞ (4.5.8) 105 Proof. A direct computation using (4.5.1) and (4.5.3) gives ∞ n ˜n{N } (x) z = B n! n=0 ∞ Bn{N } (σN x + N/2) n=0 = exz etN /2 tN etN − (z/σN )n n! N , (4.5.9) where tN = z/σN . Using the Taylor expansion, we obtain etN /2 tN tN + t2N /2 + t3N /8 + O(t4N ) = etN − tN + t2N /2 + t3N /6 + O(t4N ) z2 t2N = 1− + O(tN ) = − +O 24 2N N 3/2 This together with (4.5.9) leads to limN →∞ on any compact set in C. ∞ n=0 as N → ∞. ˜n{N } (x) zn = exz−z2 /2 uniformly B n! Bibliography [1] M. Baake and R. V. Moody, Self-similarities and invariant densities for model sets, In “Algebraic Methods and Theoretical Physics”, Y. St. Aubin ed., Springer, New York, In press. [2] M. Baake and R. V. Moody, Multi-component model sets and invariant densities, In “Aperiodic ’97”, J. L. 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[63] L. F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal., 25(1994), 1433–1460. [64] Y. P. Wang and S. L. Lee, Scale-space derived from B-spline, IEEE Trans. on Pattern Analysis and Machine Intelligence, 20(1998), 1040–1055. [65] D. X. Zhou, Spectra of subdivision operators, Proc. Amer. Math. Soc., 129(2000), 191–202. [...]... r are eigenvalues of the operator Wc,α on Lp (R) In particular, for w = 1 the w results say that the spectrum of Wc,α on Lp (R) is the closed disc with center at 12 the origin and radius α1−1/p , and that all nonzero complex numbers with absolute value strictly less than 1 are its eigenvalues We remark that the spectral properties of Wc,α are reminiscent of those of the transfer operators and their... (0, ∞), then the spectrum of the subdivision operator Sa on the weighted sequence space p w is the union of a closed disc {λ ∈ C : |λ| ≤ rρp (Sa )} and a finite set, where ρp (Sa ) is the spectral radius of Sa on p As a consequence, we also give the spectral description of the transition operator In the last Chapter, it is observed that the eigenvalues of Ta on the space of compactly supported square-integrable... condition for the Lp convergence of the cascade sequence is the StrangFix condition for the initial function φ0 , i.e φ0 (0) = 1 and φ0 (2kπ) = 0 for all 0 = k ∈ Z The sufficient condition is closely related to the spectral properties of the transition operator Ta and the subdivision operator Sa on a certain finite dimensional space The spectral radius of the operator Sa on a subspace of finitely supported sequences... p −n ·) p,w(α dx × ≤ , where C1 , C2 are positive constants independent of f and n Similarly for p = ∞, we have n Wc,α f ∞,w ≤ Cαn f ∞,w(α−n ·) , where C is independent of f and n 2 Proof of Theorem 1.1.1 If (1.0.8) holds, the boundedness of Wc,α follows from Theorem 1.1.2 We shall prove the converse by contradiction The norm of the operator Wc,α on Lp (R) will be denoted by Wc,α w Lp (R) w α+2K0... and 1.2.4 To set up the proofs of Theorems 1.2.3 and 1.2.4, we need some elementary propn erties on the support of Wc,α f, the asymptotic behavior of the weights that satisfy (1.2.2) or (1.2.5), and the relationship between the norms in Lp (R) and Lp (R) w for compactly supported functions These properties follow directly from (1.0.1), (1.2.2) and (1.2.5), and the definition of the weighted space Lp (R)... function φ0 ) may determine the convergence of the cascade algorithm in various function spaces, and the smoothness of the refinable distribution [6, 20, 30, 36, 38, 39, 42, 47, 50, 58, 59, 63] For a Banach space X and an operator T on X, we shall denote the resolvent set, spectrum, the set of all eigenvalues, the residual spectrum and the spectral radius of T on X by P (T, X), σ(T, X), σe (T, X), σr... satisfies (1.0.8) w To develop the proof of Theorem 1.1.1, we shall first establish a result, which is also essential in setting up the proof of Theorem 1.2.1 in the next section Theorem 1.1.2 Let 1 ≤ p ≤ ∞, α > 1, c be a compactly supported function in L1 (R) with R c(x)dx = 1, and w be a weight function that satisfies (1.0.7) Then there exists a positive constant C independent of n and f such that n Wc,α f... Wc,α (c − φ) Since R (1.1.6) (c(x)−φ(x))dx = 0 and supp(c−φ) ⊂ [−αK0 , αK0 ], by (1.0.6), (1.1.6) and the definition of spectral radius, there exists a positive constant C independent of 14 n such that φn − φ 1 n−1 = Wc,α (c − φ) 1 n−1 1 + α−1 2 ≤C c−φ 1 for all n ≥ 1 This gives (1.1.5) Proof of Theorem 1.1.2 For 1 ≤ p < ∞, it follows from (1.0.7), (1.1.2) and Lemma 1.1.3 that for any f ∈ Lp −n ·) (R),... eigenvalues of Ta on the space of finitely supported sequences if and only if the corresponding scaling function is a uniform B-spline It is also observed that on spaces where Ta and its adjoint share the common set of eigenvalues {2−n : n ∈ Z+ }, the corresponding eigenfunctions form a biorthogonal system comprising the distributional derivatives of the scaling function φ and an Appell sequence of polynomials... derivatives of both sides of (1.0.2) Set Σ0 := {α−k : k = 0, 1, } Then any λ ∈ Σ0 is an eigenvalue of the operator Wc,α on the Banach space Lp ([−K, K]), and any λ ∈ Σ0 \{1} is an eigenvalue of the operator Wc,α on Lp ([−K, K]) Moreover, the operator Wc,α is a compact operator on Lp ([−K, K]) 0 and on Lp ([−K, K]) for any K ≥ K0 (see [40]) Therefore, the following result 0 about spectrum of the restricted . SPECTRAL PROPERTIES OF TRANSFER OPERATORS Gao Xiaojie (M.Sc., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements I. studying the operators in weighted spaces, we bridge the gaps present among the existing results on the spectral description of the operators. We also study other spectral properties of the operators. . trace properties of the operators are characterized with the known results being listed as special cases. A set of conjugate eigensystem of adjoint operators is set up as a byproduct of the spectral