Paper3.dvi CUBO A Mathematical Journal Vol.12, No¯ 03, (203–212). October 2010 Analytic Continuation and Applications of Eigenvalues of Daubechies’ Localization Operator KUNIO YOSHINO Department of Natural Sciences, Faculty of Knowledge Engineering, Tokyo City University, Tokyo 158-8557, Japan email: yoshinok@tcu.ac.jp ABSTRACT In this paper we introduce generating functions of eigenvalues of Daubechies’ localization operator, study their analytic properties and give analytic continuation of these eigenval- ues. Making use of generating functions, we establish a reconstruction formula of symbol functions of Daubechies’ localization operator with rotational invariant symbols. RESUMEN Introducimos funciones generadas por los autovalores del operador de localización de Daubechies, estudiamos sus propiedades analíticas y damos continuación analítica de los autovalores. Haciendo uso de las funciones generadas, establecemos la fórmula de recon- strucción de funciones símbolo del operador de localización de Daubechies con símbolos rotacional invariante. Key words and phrases: Hermite functions, Daubechies (localization) operator, Borel trans- form, asymptotic expansion. Math. Subj. Class.: 33, 44, 46F. 204 Kunio Yoshino CUBO 12, 3 (2010) 1 Introduction The Daubechies (localization) operator was introduced by Ingrid Daubechies in [4], where she mainly treated localization operators with rotational invariant symbols. In particular, she expressed eigenvalues as Mellin transforms of symbol functions. She also proved that Her- mite functions are eigenfunctions of localization operators with rotational invariant symbols. So far, the theory of localization operators has been studied by several researchers in various fields ([2], [5], [7], [10], [11], [12]). In this paper we will study analytic properties of generating functions of eigenvalues of Daubechies’ localization operators. We will also give an analytic continuation of eigenvalues of Daubechies’ localization operator. Making use of generating functions, we will establish the reconstruction formula of symbol functions of Daubechies’ localization operators with ro- tational invariant symbol. For the simplicity, we will confine ourselves to the 1-dimensional case in this paper. In section 2 we will introduce Daubechies’ localization operator. In section 3 we will give the analytic continuation of eigenvalues of Daubechies’ localization operator. In section 4 we will define the generating function of eigenvalues. In final section 5 we will establish the reconstruction formulas for rotational invariant symbol function. 2 Daubechies’ Localization Operator According to [4], we define the localization operator PF as follows. Definition 1 ([4]). Daubechies’ localization operator is PF ( f )(x) = (2π)−1 ˆ ˆ R2 F( p, q)φp,q (x) < φp,q , f > d pd q, (1) where F( p, q) ∈ L1(R2), f (x) ∈ L2(R), φp,q(x) = π−1/4 ei px e−(x−q) 2/2, and < φp,q , f > denotes the inner product ˆ R φp,q(x) f (x) dx. The function F( p, q) is called the symbol function of the operator PF . Daubechies obtained the following results. CUBO 12, 3 (2010) Eigenvalues of Daubechies’ Localization Operator 205 Proposition 1 ([4]). Suppose that F( p, q) ∈ L1(R2). Then (i) If F( p, q) ≥ 0, then PF is a positive operator. (ii) PF is bounded operator. that is, ||PF ( f )||L2 ≤ (2π) −1/2||f ||L2 ||F||L1 , ( f ∈ L 2(R)). (iii) PF is a trace class operator. Proposition 2 ([4]). Suppose F( p, q) = F̃(r2), where r2 = p2 + q2. Then (i) The Hermite functions hm (x) are eigenfunctions of the operator PF : PF (hm )(x) = λm hm (x), m ∈ N. (ii) Secondly, λm = 1 m! ˆ ∞ 0 e −s s m F̃(2s)ds, m ∈ N, where the Hermite functions hm (x) are defined by hm (x) = (−1)m(2m m! p π)−1/2 exp(x2/2) dm dxm exp(−x2), m ∈ N. For details on Hermite functions, we refer the reader to [6, 7, 10]. In what follows we assume that (i) F( p, q) ∈ L1(R2). (ii) F( p, q) is rotational invariant, that is, F( p, q) = F̃(r2), where r2 = p2 + q2. 3 Analytic Continuation of Eigenvalues of Daubechies Operator In this section we consider the analytic continuation of the eigenvalues {λm } ∞ m=0. By Proposi- tion 2 we have λm = 1 m! ˆ ∞ 0 e −s s m F̃(2s)ds. 206 Kunio Yoshino CUBO 12, 3 (2010) We put λ(z) = 1 Γ(z + 1) ˆ ∞ 0 e −s s z F̃(2s)ds, z ∈ C, Re(z) > 0. where Γ(z) is Euler’s Gamma function. Then we have the following proposition. Proposition 3. λ(z) have following properties: (i) λ(z) is holomorphic in the right half plane Re(z) > 0. (ii) There exists a positive constant C such that |λ(z)| ≤ C p |z| e π 2 |y|, z = x + i y ∈ C, x > 0. (iii) λ(z) interpolates the eigenvalues {λm } ∞ m=0, that is, λ(m) = λm, m ∈ N. (iv) There exists a positive constant C such that |λm| ≤ C p |m| , m ∈ N. Proof. The proof is as follows. (i) We can prove the holomorphicity of λ(z) by Morea’s theorem and Lebesgue’s dominated convergence theorem. (iii) is obvious. (iv) follows from (ii) and (iii). So we will prove statement (ii). By Stirling’s formula, Γ(z + 1) ∼ zz e−z p 2πz, Re(z) > 0 CUBO 12, 3 (2010) Eigenvalues of Daubechies’ Localization Operator 207 and e−s sx ≤ e−x xx for s ≥ 0, we have |λ(z)| = ∣ ∣ ∣ ∣ 1 Γ(z + 1) ˆ ∞ 0 e −s s z F̃(2s) ds ∣ ∣ ∣ ∣ ≤ 1 |Γ(z + 1)| ˆ ∞ 0 e −s|sz||F̃ (2s)| ds ≤ C |zz e−z| p 2π|z| ˆ ∞ 0 e −s s x|F̃(2s)| ds ≤ C e yarg(z) xx e−x p 2π|z| ˆ ∞ 0 e −x x x|F̃(2s)| ds ≤ C p 2π|z| e π 2 |y| ˆ ∞ 0 |F̃ (2s)| ds ≤ C′ p |z| e π 2 |y|. Remark 1. The function λ(z) is the unique analytic continuation of eigenvalues {λm } ∞ m=0 be- cause of (ii) in Proproposition 3 and Carlson’s theorem [3]. 4 Generating Functions of Eigenvalues of Daubechies Op- erator In this section we introduce two generating functions Λ(w) and G(t) of the eigenvalues {λm } ∞ m=0. We begin with Λ(w). Put Λ(w) = ∞ ∑ m=0 λm w m, (|w| < 1). Due to (iv) in Proposition 3, the right-hand side is a convergent series if |w| < 1. We will show some analytic properties of Λ(w). Proposition 4. Suppose that {λm} ∞ m=0 are eigenvalues of PF . Then (i) The function Λ(w) is given by the integral Λ(w) = ˆ ∞ 0 e −s(1−w) F̃(2s)ds, Re(w) < 1. (ii) Λ(w) is holomorphic in the left-half plane {w ∈ C : Re(w) < 1} and is bounded in its closure {w ∈ C : Re(w) ≤ 1}. (iii) Λ(iv) ∈ C0(R) for v ∈ R, that is, Λ(iv) ∈ C(R) and lim|v|→∞ Λ(iv) = 0. 208 Kunio Yoshino CUBO 12, 3 (2010) Proof. We prove the three parts. (i) By (ii) in Proposition 2, we have Λ(w) = ∞ ∑ m=0 λm w m = ∞ ∑ m=0 wm m! ˆ ∞ 0 e −s s m F̃(2s) ds = ˆ ∞ 0 e −s F̃(2s) ∞ ∑ m=0 (ws)m m! ds = ˆ ∞ 0 e −s(1−w) F̃(2s) ds. (ii) For Re(w) ≤ 1, we have |Λ(w)| ≤ ˆ ∞ 0 |e−s(1−w)||F̃(2s)| ds ≤ ˆ ∞ 0 |F̃ (2s)| ds = ||F̃||L1 . (iii) Λ(iv) is the Fourier transform of the L1 function e−s F̃(2s), for s ≥ 0. Hence it belongs to C0(R n) by the Riemann–Lebesgue theorem [9]. Proposition 5. Suppose that F( p, q) is positive. If lim sup m→∞ λm 1/m = 1, then w = 1 is a singular point of Λ(w). Proof. Since F( p, q) is positive, then PF is a positive operator by Proposition 1. Therefore, all the eigenvalues of PF are nonnegative. By the Cauchy–Hadamard formula, the radius of convergence of the power series ∞ ∑ m=0 λm w m is 1. By Vivanti’s theorem, w = 1 is a singular point of Λ(w). Proposition 6. Suppose the support of F̃(2s) is contained in [0, a]. Then there exists a positive constant C such that (i) |λm| ≤ C am m! , m ∈ N. (ii) Λ(w) is an entire function of exponential type. Proof. We prove the two parts of the proposition. CUBO 12, 3 (2010) Eigenvalues of Daubechies’ Localization Operator 209 (i) Since the support of F̃(2s) is contained in the closed interval [0, a], by (ii) in Proposi- tion 2, we have λm = 1 m! ˆ a 0 e −s F̃(2s)sm ds ≤ am m! ˆ a 0 |F̃(2s)| ds. Therefore, the inequality |λm| ≤ C am m! is valid. (ii) Since |Λ(w)| ≤ ˆ a 0 |F̃(2s)|e−s(1−u) ds ≤ ea|u| ˆ a 0 |F̃(2s)|ds, we have |Λ(w)| ≤ C ea|u|, w = u + iv ∈ C. Now we consider following formal power series: ∞ ∑ m=0 m!λm t −m−1. In general, the series on right-hand side is divergent. But this formal power series is an asymptotic expansion of the Hilbert transform of F̃ (2s)e−s. Namely, if we put G(t) = ˆ ∞ 0 F̃(2s)e−s t − s ds, t ∈ C\[0,∞], then G(t) has following properties. Proposition 7. For the function G(t) we have (i) G(t) is Laplace transform of Λ(w). (ii) ∞ ∑ m=0 m!λm t −m−1 is an asymptotic expansion of G(t). Proof. 210 Kunio Yoshino CUBO 12, 3 (2010) (i) By (ii) in Proposition 4, Λ(w) is bounded in left-half plane. So we can consider the Laplace transform of Λ(w) along the negative real axis: ˆ −∞ 0 Λ(w)e−tw dw = ˆ −∞ 0 { ˆ ∞ 0 e −s(1−w) F̃(2s) ds } e −tw dw = ˆ ∞ 0 F̃(2s) e−s { ˆ −∞ 0 e w(s−t) dw } ds = ˆ ∞ 0 F̃(2s)e−s t − s ds = G(t), for Re t < 0. (ii) Secondly, G(t) = ˆ ∞ 0 F̃ (2s)e−s t − s ds = 1 t ˆ ∞ 0 F̃(2s)e−s 1 − st−1 ds = 1 t ˆ ∞ 0 F̃ (2s)e−s { N ∑ m=0 (st−1 )m + (st−1 )N+1 1 − st−1 } ds = N ∑ m=0 m!λm t −m−1 + 1 tN+1 ˆ ∞ 0 F̃(2s)e−s sN+1 t − s ds. Hence if |t| ≥ R and 0 < δ ≤ arg(t) ≤ 2π−δ, then we have |G(t) − N ∑ m=0 m!λm t −m−1| ≤ (N + 1)! R sin δ λN+1 |t|N+1 ≤ C N! p N + 1 R sin δ|t|N+1 . Proposition 8. Suppose that support of F̃(2s) is contained in [0, a]. Then (i) G(t) is holomorphic in C\[0, a]. (ii) ∞ ∑ m=0 m!λm t −m−1 converges in |t| > a. Proof. (i) From the assumption on the support of F̃(2s), we have G(t) = ˆ a 0 F̃(2s)e−s t − s ds. So G(t) is holomorphic in C\[0, a]. CUBO 12, 3 (2010) Eigenvalues of Daubechies’ Localization Operator 211 (i) By (i) in Proposition 6, |λm| ≤ C am m! , m ∈ N. Hence ∞ ∑ m=0 m!λm t −m−1 converges if |t| > a. Remark 2. The function Λ(w) is the Borel transform of G(t). For details on the Borel and Hilbert transforms, we refer the reader to [1, 8, 9]. 5 Reconstruction of Symbol Functions In this section we establish our main results. Theorem 1. The function F̃(2s) = (2π)−1 esF(Λ(iv))(s), is valid in distribution sense, where F(Λ(iv)) = ˆ ∞ −∞ e −isv) Λ(iv)dv is Fourier transform of Λ(iv). Proof. By (i) in Proposition 4, we have Λ(iv) = ˆ ∞ 0 e −s(1−iv) F̃(2s) ds = ˆ ∞ 0 e isv e −s F̃(2s) ds. This means that Λ(iv) is the inverse Fourier transform of e−s F̃(2s). Since F̃(2s) is an L1- function, then e−s F̃(2s) is a tempered distribution. Hence, as tempered distribution, we have F̃(2s) = esF(Λ(iv))(s). Theorem 2. The function F̃(2s) is given by the formula F̃ (2s) = es lim t→0 −1 2πi (G(s + it) − G(s − it)) . 212 Kunio Yoshino CUBO 12, 3 (2010) Proof. It is well known that the boundary value lim t→0 −1 2πi [G(s + it) − G(s − it)] is the inverse map of the Hilbert transform [8]. Since G(t) is Hilbert transform of e−s F̃(2s), we have lim t→0 −1 2πi (G(s + it) − G(s − it)) = e−s F̃(2s). References [1] ANDERSON, M., Topics in Complex Analysis, Springer-Verlag, New York, Berlin, Heidel- berg, 1996. [2] ASHINO, R., BOGGIATTO, P. AND WONG, M.W., Advanced in Pseudo-Differential Oper- ators, Birkhäuser-Verlag, Basel, Berlin, Boston, 2000. [3] BOAS, R.P., Entire Functions, Academic Press, New York, 1954. [4] DAUBECHIES, I., A time frequency localization operator; A geometric phase space ap- proach, IEEE. Trans. Inform. Theory, vol.34, pp. 605–612, 1988. [5] DAUBECHIES, I., Ten Lectures on Wavelets, Rutgers University and AT & T Bell Labo- ratories, 1992. [6] FOLLAND, G.B., Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. [7] GRÖHENIG, K., Foundations of Time-Frequency Analysis, Birkhäuser-Verlag, Basel, Berlin, Boston, 2000. [8] MORIMOTO, M., An Introduction to Sato’s Hyperfunction, Translation of Math. Mono- graph, vol. 129, Amer. Math. Soc. Providence, Rhode Island, 1993. [9] RUDIN, W., Real and Complex Analysis, McGraw-Hill Book Company, New York, 1987. [10] WONG, M.W., Weyl Transforms, Springer-Verlag, New York, 1998. [11] WONG, M.W., Localization Operators on the Weyl-Heisenberg Group, Geometry, Anal- ysis and Applications, Proceedings of the International Conference (editor:P.S.Pathak), pp. 303–314, 2001. [12] WONG, M.W., Wavelet Transforms and Localization Operator, Birkhäuser-Verlag, Basel, Berlin, Boston, 2002.