Time-Frequency Analysis with Wavelet Localization Operators

Time-frequency analysis for two-wawelet localization operators associated to the windowed linear canonical transform (WLCT) VIOREL CATANA University POLITEHNICA of Bucharest Department of Mathematics-Informatics catana_viorel@yahoo.co.uk THE 28TH COFERENCE ON APPLIED AND INDUSTRIAL MATHEMATICS CAIM 2021 17TH-18TH September 2021

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 5. COMPACTNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 6. REFERENCES

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALISATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 5. REFERENCES

REFERENCES 1. I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inf. Theory, 34(4) (1988), 605 612. 2. P. Boggiato and M. W. Wong, Two-wavelet localization operators on ??( ?) for the Weyl- Heisenberg group, Integr. Equ. Oper. Theory, 49 (2004), 1 10. 3. M. W. Wong, Localization operators on the affine group and paracommutators, Progress in Analysis, World Scientific, (2003), 663 669. 4. A. Calder n, Intermediate spaces and interpolation, the complex method, Studia Math. 24(2) (1964), 113 190. 5. M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136, Springer, Berlin, (2002). 6. K. Hleili and M. Hleili, Time-frequency analysis of localization operators for non-isotropic n- dimensional modified Stockwell transform, JPDOA, 12 Article number:34 (2021).

REFERENCES H. Mejjaolli and Kh. Trim che, Boundedness and compactness of localization operators associated with spherical mean Wigner transform, Complex Anal. Oper. Theory, 13(2019), 753 780. 7. C. Baccar, N. Hamadi, H. Herch, Time-frequency analysis of localization operators associated to the windowed Hankel transform., Integr. Transforms Spec. Funct. 27(3) (2016), 245 258. 8. V. Catan , Schatten-von Neumann norm inequalities for two-wavelet localization operators, In: Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, L. Rodino, B. W. Schultze, M. W. Wong, editors, AMS and Fields Institute Communications, 2007, 265 277. 9. 10. V. Catan , Two-wavelet localization operators on homogeneous spaces and their traces, Integr. Equ. Oper. Theory, 62 (2008), 351 363.

REFERENCES 11. M. Moshinsky and C. Quesnee, Linear canonical transform and their unitary representations, J. Math. Phys., 12(8) (1971), 1772 1780 12. K. Gr chenig, Foundations of Time-Frequency Analysis, Birkh user, Boston, 2001 13. M. Bahri and R. Ashino, Some properties of windowed linear canonical transform and its logarithmic uncertainly principle, International Journal of Wavelets, Multiresolution and Information Processing, 14(3) (2016), 1650015 (21 pages). 14. K. I. Kou and R. H. Xu, Windowed linear canonical transform and its applications, Signal Process, 92 (1) (2012), 179 188.

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 5. REFERENCES

INTRODUCTION The linear canonical transform (LCT) was first introduced in the early 1970s by Moshinsky and Quesnee in and is a four - parameter class of linear integral transforms. It was considered as one of the most powerful tools for signals and images processing. The LCT is a generalization of many integral transforms such as the Fourier transform (FT), the fractional Fourier transform (FRFT), the Fresnel transform, the Lorentz transform and scaling operations. The LCT has found many applications in time-frequency analysis, in filter design, signal synthesis, radar analysis, holographic three-dimensional television, quantum physics and many more. The classical windowed Fourier transform (WFT) provides simultaneously information in the time and frequency domain. It is also called the short-time Fourier transform (STFT) or Gabor transform (GT) when we use a Gaussian function.

INTRODUCTION The windowed linear canonical transform (WLCT) is a generalization of the WFT, which is constructed by replacing FT kernel with the LCT kernel in the definition of the WFT. The theory of localization operators has been initiated by Daubechies and developed by Wong and many others who have investigated this kind of linear operators in different settings. Our main aim in this paper is to introduce and study the two-wavelet localization operators related to the windowed linear canonical transform (WLCT). More precisely, we investigate their boundedness, compactness and Schatten - von Neumann properties.

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALISATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALISATION OPERATORS ASSOCIATED WITH WLCT 5. REFERENCES

PRELIMINARY A. THE LINEAR CANONICAL TRANSFORM (LCT) B. THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT)

PRELIMINARY A. THE LINEAR CANONICAL TRANSFORM (LCT) B. THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT)

THE LINEAR CANONICAL TRANSFORM (LCT) Definition (LCT) Let A = (a, b, c, d) 2 2be a real matrix parameter satisfying det A = ad bc = 1 (that is A ia a unimodular matrix). The linear canonical transform (LCT) of a signal ? ?1( ) is defined by ?(?)??(?,?)??, if ? 0 ????? (1) ??(?)(?) = 2?2?(??), if ? = 0, where ? 2 ? ??2 2 ???+? ??2, (?,?) 2 1 (2) ??(?,?) = 2???? is so-called the kernel of the LCT.

THE LINEAR CANONICAL TRANSFORM (LCT) Remark Let us remark that the parameters a, b, c, d may be real or complex numbers but we shall restrict ourselves to the case of real parameters in which case the LCT is a unitary operator on ?2( ); It is easy observe that when b = 0 the LCT of a signal is essentially a chirp multiplication. So, in this work we allways assume iii. As a special case, when A = (0,1 1,0) the LCT defined by (1) reduces to the classical Fourier transform. For the parameter matrix the LCT yields the FRFT. For the parameter matrix the LCT reduces to the Fresnel transform. i. ii. b 0; = = (cos ,sin sin ,cos ), (1, ,0,1), b b A A n , 0,

THE LINEAR CANONICAL TRANSFORM (LCT) A part of the fundamental properties of the LCT are as following: Additive property of LCT for a suitable signal f, where are real matrices and A2A1is the matrix multiplication (Kou and Xu in have called this property, the additive property of LCT). According to the additive property, the inverse transform of LCT can be defined by (3) Thus, the inverse LCT of a signal ? ?1( ) is given by ?? where if is a real 2 2 matrix and . (5) 2 1 ( , ) 2 bi i. ( ) = ( )( ) ( ) f L L f L = = , , , , 1,2 A a b c d i A A 2 1 A A i i i i i 1 2 = 1( )( ) L L f f A A 1(?)(?) = ?(?)?? 1(?,?)?? (4), 1 ( , , , ) A d b c a = = ( , , , ) a b c d A i a 2 b d b 2 2 + x x = = b ( , ) x K x e K A 1 A

THE LINEAR CANONICAL TRANSFORM (LCT) The equivalent way to write the inversion property is: ? 2 ? ??2 2 ??? ? ??2 1 1??(?) (?) = ?(?) = ?? ?? = ??(?)(?)?? 1(?,?) ?? (6). 2??? ??(?)(?)? The Parseval formula of LCT ?(?)?(?) ?? = (?,?) = ??(?), ??(?) = ??(?)(?)??(?)(?) ??,for all ?, ? ?2( ).(7) ii. The Plancherel formula of LCT 2 2 ( ) A f iii. (8) 2 f L 2 = The asymptotic property = iv. lim ( )( ) f 0; L A

PRELIMINARY A. THE LINEAR CANONICAL TRANSFORM (LCT) B. THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT)

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) Definition (WFT) Let ? ?2( )\ 0 be a window function. Then its window daughter function or its windowed Fourier kernel ??,?: is defined by ??,?(?) = ?(? ?)????, ? (9). The WFT of a signal ? ?2( ) with respect to the window function ? ?2( )\ 0 is defined by ??? ?,? = ? ? ??,??? = ?,??,? (10).

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) Definition (WLCT) Let ? ?2( )\ 0 be a window function. Then its window daughter function associated with LCT or the linear canonical windowed Fourier kernel is defined by ??,? (11). ?(?) = ??(?,?) ?(? ?), ? Then the WLCT of a signal ? ?2( ) with respect to the window function ? ?2( )\ 0 is defined by ?? ?? = ?(?)?(? ?)??(?,?) ?? == ??(????)(?) = 1 2??? ?(?)?(? ?) ??? ?(?)(?,?) = ?(?)??,? ? ? 2 ? ??2 2 ???+? ??2 (12).

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) Remark (i) The WLCT of a signal? ?1( ) with respect to a window function ? ? ( ) can also be defined in the same way as in (12); (ii) Similarly, we can define by (12) the WLCT in the case ? ??( ) and ? ??( ), 1 ?, ? 1 ?+1 ?(? ?), ? ). ?= 1 (because by Holder s inequality it follows that ???? ?1( ), where ???(?) = ,

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) A part of the fundamental properties of the WLCT are as following: Orthogonality relation of WLCT Let f, g, , ?2(R), 0, 0. Then ?? i. ( ) A A = ?(?) ?2( ) and (Parseval s formula) (13). ( ), f ( ) g ( , )( , ) f g G G ?(?), ?? A = (Plancherel s formula) (14) ( ) f G 1, f If ?,? ?2( ), then In particular, if then 2 2 2 = 2 A = ( ) f G f (15) 2 2 Thus, the WLCT is an isometry from ?2( ) to ?2( ).

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) A part of the fundamental properties of the WLCT are as following: Inversion formula for WLCT ii. Let? ?2( ) and ?,? ?2( ), (?,?) 0.Then 1 (?,?) ?? ?(?)(?,?)??,? ?(?)????, ? (16). ?(?) =

THE WINDOWED LINEAR CANONICAL TRANSFORM (WLCT) Some the auxiliary results that will be used in the proof of the next theorems concerning the boundedness of the two-wavelet localization operators associated with WLCT are as following: Lemma 1 Let ?,? ?2( ) be two window functions. Then, for every and ?,? ?2( ) the function belongs to ??( ) and ( ) ( ) G f G g 1 p A A ( ) ( ) G f G g 1 p 1 1 A A f g (17). p 2 2 2 2 2 b Lemma 2 1 p Let and q be the conjugate exponent of p. Then, for all window functions ? ??( ), ? ??( ) and for every function ? ??( ), we have 1 ( )( , ) 2 b A A , (?,?) 2. (18) , u p G f u f p p q

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 5. REFERENCES

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Definition Let ? ?1( 2) be a symbol, ? ?1( )\ 0 and ? ? ( )\ 0 be two window functions. Then, the two-wavelet localization operator associated with WLCT ??,?,?:? ( ) ? ( ) is defined by 1 (?,?) ?(?,?)?? ? ? ( ), ? ( ). ?(?)(?,?)??,? ?(?)????, (19) ??,?,?(?)(?) = Let us remark that it follows from relation (19) that the function ??,?,?(?) ? ( ). In addition, we have 1 1 ?1? ?1? , ? . (20) ??,?,?(?)(?) 2? ?, (?,?)

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Theorem 1 1 ?+1 ( , ) Let ? ?1( 2),? ??( ), ? ??( ), 1 ?, ? , two-wavelet localization operator ??,?,?:??( ) ??( ) is bounded and in addition we have the following estimation of the operator norm: ?= 1 such that Then, the 0. 1 1 (21). ??,?,? ?(??( )) ?1???? (?,?) 2? ? The proof of this theorem relies on Lemma 2 and the Minkowski integral inequality.

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Corollary Let ? ?1( 2) and ?,? ??1( ) ??2( ) be two window functions, 1 q and q2 is the 1 1 p 1 = + . (0,1) q q conjugate exponent of q1. For we define p by Then, the two-wavelet localization operator ??,?,?:??( ) ??( ) is bounded and the following estimate is valid: (22) ( 2 1 1 1 ( ( )) ( , ) 2 b 1 2 ) ( ) 1 1 1 L , , q q p q q B 2 L The proof of the above corollary follows by using the Riesz-Thorin interpolation Theorem and the Theorem 1. The following two propositions give auxiliary results which are needed in the proof of the next theorems.

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Proposition 1 Let ? ??( 2), 1 ? , ?,? ?2( )\ 0 be two window functions such that Then, the two-wavelet localization operator ??,?,?:?2( ) ?2( ) satisfies the following relation: 1 (?,?) ?(?,?)?? ( , ) 0. ?(?)(?,?)?? ?(?)(?,?) ???? (23). ??,?,?(?),? = 1 ????2?2 1 1 Moreover, (24). ??,?,? ?(??( )) (?,?) 2? ? Proposition 2 Let ? ?1( 2), ? ??( ), ? ??( ) be two window functions such that where and q is the conjugate exponent of p. Then, the adjoint of the two-wavelet localization operator ??,?,?:??( ) ??( ) is ??,?,?:??( ) ??( ). ( , ) 0, 1 , p q

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Theorem 2 Let ? ??( 2), 1 ? be a symbol and ? ?2( ) ? ( ), ? ?1( ) ?2( ) be two window functions such that Then, the two-wavelet localization operator 2? ?+1( ) ? 1 ? ? ?1 ( , ) 0. 2? ?+1( ) is a bounded linear operator such that ??,?,?:? 1 ? 1 1 ? 1 1 (25) ??,?,? ?2?2 ?? 2? ?+1( )) (?,?) 2? ? ?(? The proof of Theorem 2 follows by applying the Theorem 1 and by using Calderon s multi-linear interpolation theory

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Remark 2? ?+1( ), for it follows that 1 p By using Proposition 3.4 and be inclusion ?1( ) ?2( ) ? Theorem 2 asserts in fact that the two-wavelet localization operator ??,?,?:?1( ) ?2( ) ?2( ) can be uniquely extended to a bounded linear operator from ? Theorem 3 Let ? ??( 2), 1 ? be a symbol and ? ?1( ) ?2( ), ? ?2( ) ? ( ) be two window functions such that Then the two-walvelet localization operator ??,?,?:?1( ) ?2( ) ?2( ) can be uniquely extended to a bounded linear operator from ? 1 ? ?1? 2? ?+1( ) into itself. ( , ) 0. 2? ? 1( ) into itself such that 1 ? 1 1 ? 1 1 (26) ??,?,? ?2?2 ?? 2? ? 1( )) (?,?) 2? ? ?(? The proof of Theorem 3 follows by using Proposition 2 and by applying Theorem 2.

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT By combining Theorems 2, 3 and by applying the Riesz-Thorin interpolation Theorem we can state the following result. Theorem 4 Let ? ??( 2), 1 ? be a symbol and ?,? ?1( ) ?2( ) ? ( ) be two window functions such that Then, the two-wavelet localization operators ??,?,?:? 2? ? 1( ) can be uniquely extended to a bounded linear operator from ??( ) to ??( ), where Moreover, 2? ?+1( ) ? 2? ? 1( ) ? 2? ?+1 2 1 p + ( , ) 0. 2 p p p ? . q 1 1 ? 1 ? ? ? ? ? 1 ? 1 1 ??, (27) ??,?,? ?(??( )) 1 2 p p + ? ?1 ?1? ?2?2 (?,?) 2? ? t t 1 q + 1 p p q where or equivalently 2 1 1 p = + = t . p 2

BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT In the next theorem we give a result concerning the boundedness of the two-wavelet localization in the case theirs symbols belong to ??( 2) with 1 2. r Theorem 5 Let ? ??( 2), 1 ? 2 be a symbol and ?,? ?1( ) ?2( ) ? ( ) be two window functions such that Then, there exists a unique bounded linear operator ??,?,?:??( ) ??( ), where and is the conjugate exponent of r. In addition we have the following estimate of [0,1] p r r ( , ) 1 tt 0. r t 1 = + ??2 1 ???, the operator norm of the last operator: ??,?,? where 1 1 ( , ) 2 (28) ?(?2( )) ?1 + 1 r + 1 r 2 r 2 r 1 1 2 2 r 2 r r ( ) ( ) 2 r ( ) ( ) 1 1 = . K = , K 2 ( , ) 1 2 2 1 2 b 1 2 2 b

OVERVIEW 1. INTRODUCTION 2. PRELIMINARY 3. BOUNDEDNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 4. SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO- WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 5. REFERENCES

SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Given we define the Schatten-von Neumann p-class of H (H is a separable and complex Hilbert space), denoted or simply Sp, to be the space of all compact operators with its singular values sequence belonging to ??( ) (the p-summable sequences space), that is ( ) 1 k = 0 p ( ) H ( ) k s T S : T H H p 1 k p ( ) . s T k We will be concerned with the range the norm. T In this case, Spis a complex Banach space with 1 . p 1 p ( ) p = ( ) , s T T S (29). 1 k = : T H H ( 1 k = k p S p ( ) Tr T Let of T by (30) where is any orthonormal basis of H. ) ( ) , , k k tr T Tv v be a bounded linear operator in the trace class. S1. Then we can define trace = 1 k k v

SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Proposition ( ) Let ? ?1( 2) be a symbol and ?,? ?2( ) be two window functions such that , 0. Then the two-wavelet localization operator ??,?,?:?2( ) ?2( ) is in the Hilbert-Schmidt class S2and we have 1 (?,?) 2? ? ?2?2?1 1 (31). ??,?,? ?2

SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Theorem 1 ( ) Let ? ?1( 2) be a symbol and ?,? ?2( ) be two window functions such that Then the two-wavelet localization operator ??,?,?:?2( ) ?2( ) belongs to the trace class S1 and we have 1 (?,?) ?(?,?) ??,? , 0. ?,??,? ? ???? (32). ??(??,?,?) = Theorem 2 ( ) Let ? ??( 2), 1 ? and ?,? ?2( ) such that Then the two-wavelet localization operator ??,?,?:?2( ) ?2( ) belongs to the Schatten-von Neumann class Sp. , 0. 1 ??2?2?? 1 1 Moreover, we have ??,?,? (33). ?? (?,?) 2? ?

SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Theorem 3 1 p 1 q ( ) + = 1 , 0. Let ? ??( 2), ? ??( 2), 1 ?,? < , and ?,? ?2( ) such that Then the bounded linear operator ??,?,???,?,?:?2( ) ?2( ) is in the trace class S1. In addition we have 2 1 1 2????(34) ?? ??,?,???,?,? ?2?2 (?,?) 2? ? The next theorem gives a lower bound for the trace class norm ??,?,? operator ??,?,?:?2( ) ?2( ). ?1of the two-wavelet localization

SCHATTEN-VON NEWUMANN PROPERTIES OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT Theorem 4 Let ? ?1( 2) and ?,? ?2( ) such that ?,? 0. Then the trace class norm of the two-wavelet localization operator ??,?,?:?2( ) ?2( ) satisfies the following estimate: 2 ?2 where ?(?,?) = ??,?,???,? ,??,? , (?,?) 2. 1 1 (35) ?1 ??,?,??1 ?2?2?1, 2+ ?2 2 (?,?) 2? ? ? ?

COMPACTNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT p 2 2R L Lp ( Proposition 5.1. Let functions such that ( : , , L L The proof of Proposition 5.1 follows by using Propositions 3.4, 4.1 and the fact that the limit of a sequence of compact operators in Proposition 5.2. Let ) ( L be a symbol and two window functions such that 0 ) , ( operator ) ( ) ( : , , R L R L L . The proof of Proposition 5.2 relies on Theorem 3.9 and the definition of the compactness. be a symbol, 0 . Then the two-wavelet localization operator is compact. and be two window 1 ( , ) , ( ) R ) 2 2 ) ( ) R L R 2R L is compact. ( ( )) B 1R 2 1 2 be , ( ) ( ) ( ) L R L R L R . Then there exists a unique compact linear 1 1

COMPACTNESS OF THE TWO-WAVELET LOCALIZATION OPERATORS ASSOCIATED WITH WLCT 1R L 2 1 2 Theorem 5.3. Let such that , ( for all p 1 The proof of the above theorem follows by using Theorem 3.2, Propostion 5.2 and the interpolation of compact operators. Theorem 5.4. Let ) ( R Lr , ) ( ) ( ) ( , R L R L R L be two window functions such that such that 1 ?= ) 1 , 0 ( t operator ) ( ) ( : , , R L R L L is compact. The proof of the last theorem follows by using Proposition 5.2, Theorem 4.3 and the interpolation of compact operators. and be window functions ) ( ) R L R is compact ( ) , ( ) ( L ) ( ) L R L R L : R ( p p . Then the bounded linear operator ) 0 L , , . r 2 be a , symbol 0 . Then for all p and 1 2 ( 1 2 ) ? ?+1 ? p and r is the conjugate exponent of r, the bounded linear ? , p