S. Molahajloo and M.W. Wong SQUARE-INTEGRABLE GROUP REPRESENTATIONS AND LOCALIZATION OPERATORS FOR MODIFIED STOCKWELL TRANSFORMS

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1 Rend. Sem. Mat. Univ. Pol. Torino Vol. 67, , Second Conf. Pseudo-Differential Operators S. Molahajloo and M.W. Wong SQUARE-INTEGRABLE GROUP REPRESENTATIONS AND LOCALIZATION OPERATORS FOR MODIFIED STOCKWELL TRANSFORMS Dedicated to Professor Luigi Rodino on the occasion of his 60th birthday Abstract. Recently discovered square-integrable group representations are used to study localization operators for the modified Stockwell transforms. The Schatten von Neumann properties of these localization operators are established in this paper, and for trace class localization operators, the traces and the trace class norm inequalities are presented.. Introduction Let ϕ L R L 2 R. Then as a hybrid of the Gabor transform and the wavelet transform, the Stockwell transform S ϕ f of a signal f in L 2 R with respect to the window ϕ is defined by S ϕ fb,ξ2π /2 ξ e ixξ fxϕξx bdx for all b in R and ξ in R\{0}. Alternatively, we can write for all f in L 2 R, b in R and ξ inr\{0}, S ϕ fb,ξ f,ϕ b,ξ L 2 R, where ϕ b,ξ 2π /2 M ξ T b D ξ ϕ, the modulation operator M ξ, the translation operator T b and the dilation operator D ξ are defined by M ξ hxe ixξ hx, T b hxhx b, D ξ hx ξ hξx, for all x inrand all measurable functions h onr. The Stockwell transform is a versatile tool first introduced in []. More recent results on the Stockwell transform in the contexts of applications can be found in [6, 0]. The mathematical underpinnings of Stockwell transforms are developed in [4, 5, 7, 8, 9, 3]. Canada. This research has been supported by the Natural Sciences and Engineering Research Council of 25

2 26 S. Molahajloo and M.W. Wong Prompted by applications in time-frequency analysis, the Stockwell transform S ϕ has been extended in [6, 7] to a family {Sϕ s : 0 < s } of modified Stockwell transforms, which include the classical Stockwell transform when s and a variant of the wavelet transform when s2. To wit, for all functions in L 2 R, the modified Stockwell transform Sϕ s f of f for 0<s is defined by where S s ϕ fb,ξ f,ϕb,ξ s L 2 R, b R,ξ R\{0}, ϕ b,ξ s 2π /2 M ξ T b D s ξ ϕ, and for all t in0, ] the dilation operator D t ξ is defined by D t ξ hx ξ /t hξx for all x inrand all measurable functions h onr. More explicitly, Sϕ s fb,ξ2π /2 ξ /s e ixξ fxϕξx bdx for all b inrand ξ inr\{0}. For a comparison with the classical Stockwell transform, we note that S s ϕ fb,ξ ξ /s S ϕ fb,ξ, b R,ξ R\{0}, where s is the conjugate index of s given by /s+/s. An important property of the modified Stockwell transform is the following resolution of the identity formula in [6, 7]. and THEOREM. Let ϕ L R L 2 R be such that ϕxdx ˆϕξ 2 dξ<, ξ where ˆϕ is the Fourier transform of ϕ defined by ˆϕξ2π /2 e ixξ ϕxdx, ξ R. Then for all f and g in L 2 R, we get for 0<s, where f,g L 2 R c ϕ c ϕ S s ϕ fb,ξss ϕ gb,ξ dbdξ ξ 2/s, ˆϕξ 2 dξ. ξ

3 Localization operators for modified Stockwell transforms 27 Based on the resolution of the identity formula in Theorem, localization operators can be introduced and their Schatten von Neumann properties investigated. Results in this direction are announced in [8]. The aim of this paper is to use square-integrable group representations found by Boggiatto, Fernández and Galbis [4] to construct localization operators. These localization operators turn out to be the same as the localization operators based on the resolution of the identity formulas for the modified Stockwell transforms. From this fact follow the Schatten von Neumann properties, a trace formula and trace class norm inequalities for the localization operators defined using the resolution of the identity formulas for the Stockwell transforms. In Section 2, we give a brief recapitulation of Schatten von Neumann classes and localization operators corresponding to square-integrable representations of locally compact and Hausdorff groups. Square-integrable group representations suggested by the one in [4] are summarized in Section 3. In Section 4, localization operators arising from these square-integrable representations are introduced. They are shown to coincide with localization operators defined using the resolution of the identity formulas for the modified Stockwell transforms. The Schatten von Neumann properties of these localization operators are established, and the traces of trace class localization operators are computed. We give in Section 5 the trace class norm inequalities for the trace class localization operators studied in Section 4 and give an explicit formula for the function that occurs in the lower bound for the trace norm of such a trace class localization operator. 2. Schatten von Neumann classes and localization operators Let X be an infinite-dimensional, complex and separable Hilbert space in which the inner product and norm are denoted, respectively, by, and. Let A : X X be a compact operator. Then the operator A : X X defined by A A A is positive and compact. So, using the spectral theorem, there exists for X an orthonormal basis {ϕ k : k,2,...} consisting of eigenvectors of A. For k,2,..., let s k be the eigenvalue of A : X X corresponding to the eigenvector ϕ k. We say that the compact operator A : X X is in the Schatten von Neumann class S p, p<, if k s p k <. If a compact operator A : X X is in S p, p<, then we define the norm A Sp of A by A Sp { /p sk} p. k

4 28 S. Molahajloo and M.W. Wong By convention, the Schatten von Neumann class S is taken to be simply the C - algebra BX of all bounded linear operators on X and the norm S in S is simply the norm in BX. Of particular interest is the Schatten von Neumann class S, which is also known as the trace class. If a compact operator A : X X is in the trace class S, then we can define the trace tra of A by tra k Aϕ k,ϕ k, where{ϕ k : k,2,...} is any orthonormal basis for X. Let G be a locally compact and Hausdorff group on which the left Haar measure is denoted by dµ. Let UX be the group of all unitary operators on X and let π : G UX be an irreducible and unitary representation of G on X. Suppose that the representation π is square-integrable in the sense that there exists a nonzero vector ϕ in X such that 2 ϕ,πgϕ 2 dµg<. G The condition 2 is known as the admissibility condition for the square-integrable representation of G on X. We call any vector ϕ for which ϕ and the admissibility condition 2 is fulfilled an admissible wavelet for the square-integrable representation of G on X. For any admissible wavelet ϕ, we define the constant c ϕ by c ϕ ϕ,πgϕ 2 dµg. G We need the following result, which is Theorem 4.5 in [2]. THEOREM 2. Let ϕ be an admissible wavelet for a square-integrable representation π : G UX of G on X. Let F L p G, p. For every x in X, we define L F,ϕ x in X by L F,ϕ x,y c ϕ G Fgx,πgϕπgϕ,ydµg for all y in X. Then L F,ϕ : X X is in the Schatten von Neumann class S p and L F,ϕ Sp c /p ϕ F L p G. REMARK. The linear operator L F,ϕ : X X is called the localization operator for the transform X x x,π ϕ L 2 G. The following trace formula is given in Theorem 3.6 in [2]. THEOREM 3. Let ϕ be an admissible wavelet for a square-integrable representation π : G UX of G on X. Let F L G. Then the trace trl F,ϕ of the trace

5 Localization operators for modified Stockwell transforms 29 class localization operator L F,ϕ : X X is given by trl F,ϕ c ϕ G Fgdµg. A lower bound for the norm L F,ϕ S of the trace class localization operator L F,ϕ : X X can be given in terms of the function F ϕ on G defined by F ϕ gl F,ϕ πgϕ,πgϕ, g G. Indeed, we have the following result, which is Theorem 4. in [2]. THEOREM 4. Let ϕ be an admissible wavelet for a square-integrable representation π : G UX of G on X. Let F L G. Then c ϕ F ϕ L G L F,ϕ S c ϕ F L G. The function F ϕ is the expectation value of the observable L F,ϕ : X X in the coherent states πgϕ, g G. Information about coherent states and related topics can be found in [, 2, 3]. 3. Square-integrable representations Let G be the set R R\{0} S, where S is the unit circle centered at the origin. If we identifys with the interval[ π,π], thengbecomes a group with respect to the multiplication given by b,ξ,θ b 2,ξ 2,θ 2 b + b 2,ξ ξ 2,θ + θ 2 + b ξ ξ 2 ξ for allb,ξ,θ andb 2,ξ 2,θ 2 ing. In fact,gis a Lie group on which the left Haar measure is just the Lebesgue measure. For future reference, let us note that 0,,0 is the identity element ingand b,ξ,θ bξ,/ξ, θ+b ξ for all b,ξ,θ ing. For α,, we let H α be the set defined by { } H α f S R : ˆfu 2 u α du<. Then H α becomes a Hilbert space in which the inner product, Hα Hα are given by f,g Hα ˆfuĝu u α du and the norm

6 220 S. Molahajloo and M.W. Wong and f 2 H α ˆfu 2 u α du for all f and g in H α. We assume throughout this paper that α,. Let UH α be the group of all unitary operators on H α. Then we define the mapping ρ α :G UH α by 3 ρ α b,ξ,θ f e iθ+bξ ξ α+/2 π b,ξ f for all b,ξ,θ ingand all f in H α, where π b,ξ fx ξ fξx b, x R. THEOREM 5. ρ α :G UH α is an irreducible and unitary representation of G on H α. In fact, the following theorem tells us much more about the representation ρ α : G UH α. THEOREM 6. The representation ρ α : G UH α of the group G on H α is square-integrable. THEOREM 7. Let ϕ H α, H α,, where { } H β, f S : ˆfu 2 u+ β du<, β,. Let ψ be the function onrdefined by ψt2π /2 e itw w α ˆϕw dw, t R. Then for all f in H α, the modified Stockwell transform Sϕ s f of f for 0<s, is given by f,ρ α b,ξ,θψ Hα 2π /2 e iθ ξ α+/2 /s S s ϕ fb,ξ, b,ξ,θ G. REMARK 2. In fact, ψf α ˆϕ, wheref denotes the inverse Fourier transform. THEOREM 8. Let f and g be in H α. Then for 0<s and for all ϕ H α,, Sϕ s fb,ξsϕgb,ξ s dbdξ ξ 2/s α+ ϕ 2 H α, f,g Hα. Theorem 8 can be seen as another set of resolution of the identity formulas for the modified Stockwell transforms Sϕ s, 0<s, and is the basis for the localization operators studied in the following section.

7 Localization operators for modified Stockwell transforms Localization operators for modified Stockwell transforms Let ϕ be a nonzero function in H α, H α, and let ψ be the function onrdefined by 4 ψt2π /2 e itw w α ˆϕw dw, t R. Without loss of generality, we can choose ϕ in such a way that 5 ψ Hα. Let F L p G, p. Then for all f in H α, we define L F,ψ f by L F,ψ f,g Hα π π for all g in H α, where Fb,ξ,θ f,ρ α b,ξ,θψ Hα ρ α b,ξ,θψ,g Hα dbdξdθ π π ψ,ρ α b,ξ,θψ Hα 2 dbdξdθ. LEMMA. Let ϕ and ψ be as in 4 and 5. Then Proof. We note that π By Theorem 7, and hence So, π π π π π 4π 2 ϕ 2 H α,. ψ,ρ α b,ξ,θψ Hα 2 dbdξdθ ψ,e iθ+bξ ξ α+/2 π b,ξ ψ Hα 2 dbdξdθ ψ, ξ α+/2 π b,ξ ψ Hα 2 dbdξdθ. ψ,e ibξ π b,ξ ψ Hα 2π /2 ξ α S ϕ ψb,ξ, b R, ξ R\{0}, ψ,π b,ξ ψ Hα 2π /2 e ibξ ξ α S ϕ ψb,ξ, b R, ξ R\{0}. π π 4π 2 Since ψ Hα, the lemma is proved. ψ,ρ α b,ξ,θψ Hα 2 dbdξdθ S ϕ ψb,ξ 2 dbdξ ξ α 4π2 ϕ 2 H α, ψ 2 H α.

8 222 S. Molahajloo and M.W. Wong Now, by Theorem 2, we can conclude that L F,ψ : H α H α is in the Schatten von Neumann class S p. This fact can be used to prove the following result. THEOREM 9. Let ϕ be as given in 4 and 5, F L p R R, p. If for 0<s, we define L s F,ϕ f for all f in H α by L s F,ϕ f,g H α ϕ 2 H α, Fb,ξS s ϕ fb,ξss ϕ gb,ξ dbdξ ξ 2/s α+ for all g in H α, then L s F,ϕ : H α H α is in the Schatten von Neumann class S p. Moreover, L s F,ϕ Sp 2π ϕ 2 H α, /p F L p R R. Proof. If we define the function F ongby Fb,ξ,θFb,ξ, b,ξ,θ G, then F L p G. But for all f and g in H α, we get by 3 L F,ψ f,g H α 6 2π π π By, 3 and Theorem 7, we have Fb,ξ,θ f,ρ α b,ξ,θψ Hα ρ α b,ξ,θψ,g Hα dbdξdθ Fb,ξ f,π b,ξ ψ Hα π b,ξ ψ,g Hα ξ α+ dbdξ. 7 f,π b,ξ ψ Hα 2π /2 e ibξ ξ α+ /s S s ϕ fb,ξ and 8 π b,ξ ψ,g Hα 2π /2 e ibξ ξ α+ /s S s ϕ gb,ξ for all b in R and ξ in R\{0}. Putting 7 and 8 in 6 and using Lemma, we get for all f and g in H α, 9 L F,ψ f,g H α 4π2 L s F,ϕ f,g Hα. Fb,ξS s ϕ fb,ξss ϕ gb,ξ dbdξ ξ 2/s α+ So, L s F,ϕ : H α H α is the same as L F,ϕ : H α H α and is hence in the Schatten von Neumann class S p. Finally, using the inequality in Theorem 2 and Lemma, we get L s F,ϕ Sp L F,ψ S p F L p G 2π ϕ 2 H α, /p F L p R R.

9 Localization operators for modified Stockwell transforms 223 REMARK 3. It is important to bring out the fact that by 9, the localization operators L s F,ϕ are all equal to L F,ψ for all s in 0, ]. This fact can also be seen from the formula. Notwithstanding the variety of Hilbert spaces H α offered by Theorem 9, it is to be noted, however, that in view of applications to signal analysis and imaging, the Hilbert space H 0, i.e., L 2 R, is most commonly used. A formula for traces of localization operators for the modified Stockwell transforms is given in the following theorem. THEOREM 0. Let ϕ be as given in 4 and 5. Then for all functions F in L R R, the trace trl s F,ϕ of the trace class localization operator Ls F,ϕ : H α H α, 0<s, is given by trl s F,ϕ 2π ϕ 2 H α, Proof. By Theorem 3, Lemma and 9, we get as required. trl s F,ϕ tr L F,ψ π π 2π ϕ 2 H α, Fb,ξdbdξ. Fb,ξ,θdbdξdθ Fb,ξdbdξ, 5. Trace class norm inequalities We give in this section a result on the trace class norm inequalities for the localization operators L s F,ϕ : H α H α, 0<s. THEOREM. Let ϕ be as given in 4 and 5. Then for all functions F in L R R, we get for 0<s where 2π ϕ 2 F ϕ L R R Ls F,ϕ S H α, 2π ϕ 2 F L R R, H α, F ϕ b,ξ for all b inrand ξ inr\{0}. Fb,ξ ϕ,t ξb b M ξ /ξ D 2/α+ ξ /ξ ϕ H α, 2 db dξ To prove Theorem, we note that by Theorem 9, we only need to establish the lower bound for L s F,ϕ S. To that end, let us recall that by Remark 3, L s F,ϕ L F,ψ,

10 224 S. Molahajloo and M.W. Wong where ψ is given in 4 and Fb,ξ,θFb,ξ, b,ξ,θ G. So, by Theorem 4, F ψ L G Ls F,ϕ S. By Lemma, Theorem is proved if we show that F ψ L G 2π F ϕ L R R, where F ψ b,ξ,θ L F,ψ ρ αb,ξ,θψ,ρ α b,ξ,θψ, b,ξ,θ G. H α This follows from the next formula. THEOREM 2. Under the hypotheses of Theorem, F ψ b,ξ,θ 2π Fb,ξ ϕ,t ξb b M ξ /ξ D 2/α+ ξ ϕ /ξ H α, 2 db dξ. We give two proofs of Theorem 2. The first proof is based on the explicit Fourier transform of ρ α b,ξ,θψ for allb,ξ,θ ingand the second one, in which the same Fourier transform is still a key ingredient, explicates the use of the underlying group structure. First proof of Theorem 2. The starting point is the formula u 0 ρ α b,ξ,θψ ue iθ+bξ ξ α /2 e ibu ˆϕ ξ u α, u R, for all b,ξ,θ ing. So ρ α b,ξ,θψ,ρ α b,ξ,θ ψ H α ρ α b,ξ,θψ uρ α b,ξ,θ ψ u u α du u u ξξ α /2 ˆϕ ξ ˆϕ ξ e ib b u u α du

11 Localization operators for modified Stockwell transforms 225 ξξ α /2 ξv+ ξ ˆϕv ˆϕ e ib b ξv v+ α ξ α dv ξ ξ α /2 ξ ˆϕue ib b ξv v+ ξ ˆϕ /ξ ξ v+ α dv /ξ ξ α /2 ξ ˆϕu ξ ξ T b b ξm ξ /ξ D ξ /ξϕ v v+ α dv ˆϕvT b b ξm ξ /ξ D 2/α+ ξ /ξ ϕ,t b b ξm ξ /ξ D 2/α+ ξ ϕ /ξ for all b,ξ,θ ing. Hence F ψ b,ξ,θ 2π π π H α, ϕ v v+ α dv Fb,ξ ρ α b,ξ,θψ,ρ α b,ξ,θ ψ Hα 2 db dξ dθ Fb,ξ for all b,ξ,θ ing, as claimed. ϕ,t ξb b M ξ /ξ D 2/α+ ξ /ξ ϕ H α, 2 db dξ Second proof of Theorem 2. Since ρ α : G UH α is a unitary representation of G on H α, it follows that ρα b,ξ,θψ,ρ α b,ξ,θ ψ H α ψ,ρ α b,ξ,θ b,ξ,θ ψ Hα ψ,ρ α ξb b,ξ /ξ,θ θ+bξ ξψ Hα. To simplify notation, we let bξb b, ξξ /ξ and θθ θ+bξ ξ. Then by 0, ρα b,ξ,θψ,ρ α b,ξ,θ ψ H α ψ,ρ α b, ξ, θψ H α ˆψuρ α b, ξ, θψ u u α du ũ u α ˆϕu e i θ+ b ξ e i bu ξ α /2 ˆϕ ξ u α u α du

12 226 S. Molahajloo and M.W. Wong e i θ+ b ξ ξ α /2 e i b e i θ+ b ξ ξ α /2 ũ ˆϕu e i bu ˆϕ ξ u α du e i b e i θ+ b ξ ξ α /2 e i b e i θ+ b ξ ξ α+/2 ˆϕve i bv ˆϕ e i b e i θ+ b ξ ϕ,t b M ξ D 2/α+ ϕ ξ H α, and then we can proceed as in the first proof. v+ ξ v+ ξ α dv ˆϕvM b T ξ D / ξ ˆϕv v+ α dv ˆϕvT b M ξ D ξϕ v v+ α dv References [] ALI S.T., ANTOINE J.-P. AND GAZEAU J.-P., Coherent States, Wavelets and Their Generalizations, Springer-Verlag, [2] ALI S.T., ANTOINE J.-P., GAZEAU J.-P. AND MUELLER U.A., Coherent states and their generalizations: A mathematical overview, Rev. Math. Phys , [3] BEREZIN F.A. AND SHUBIN M.A., The Schrödinger Equation, Kluwer Academic Publishers, 99. [4] BOGGIATTO P., FERNÁNDEZ C. AND GALBIS A., A group representation related to the Stockwell transform, Indiana Univ. Math. J., to appear. [5] DU J., WONG M.W. AND ZHU H., Continuous and discrete inversion formulas for the Stockwell transform, Integral Transforms Spec. Funct , [6] GUO Q., MOLAHAJLOO S. AND WONG M.W., Modified Stockwell transforms and time-frequency analysis, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 89, Birkhäuser, 2009, [7] GUO Q. AND WONG M.W., Modified Stockwell transforms, Acc. Sc. Torino - Memorie Sc. Fis., Mat. e Nat., serie V, , [8] LIU Y., Localization operators for two-dimensional Stockwell transforms, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 89, Birkhäuser, 2009, [9] LIU Y. AND WONG M.W., Inversion formulas for two-dimensional Stockwell transforms, in Pseudo- Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Communications Series 52, American Mathematical Society, 2007, [0] STOCKWELL R.G., Why use the S-transforms?, in Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Communications Series 52, American Mathematical Society, 2007, [] STOCKWELL R.G., MANSINHA L. AND LOWE R.P., Localization of the complex spectrum, IEEE Trans. Signal Processing , [2] WONG M.W., Wavelet Transforms and Localization Operators, Birkhäuser, 2002.

13 Localization operators for modified Stockwell transforms 227 [3] WONG M.W. AND ZHU H., A characterization of the Stockwell spectrum, in Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications 72, Birkhäuser, 2007, AMS Subject Classification: 65R0, 94A2; 47G0, 47G30 S. MOLAHAJLOO, M.W. WONG, Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J P3, CANADA ÑÓÐÐ Ñ Ø Ø ØºÝÓÖ Ùº Lavoro pervenuto in redazione il ÑÛÛÓÒ Ñ Ø Ø ØºÝÓÖ Ùº