A new convolution theorem associated with the linear canonical transform

A new con v olution theorem asso ciated with the l inear canonical trans form Haiy e Huo Departmen t of Mathematics, Nanc hang Univ ersit y , Nanc hang 330031, China Email: hy h uo@ncu.edu.cn Abstr act . In this pap er , w e first in tro du ce a new notion of canonical con v olution op er- ator, and sho w that it satisfies the comm utativ e, asso ciativ e, an d distributive prop erties, whic h ma y b e quite useful in signal pro cessing. Moreo v er, it is p ro v ed that the generalized con v olution theorem and generalized Y oung’s in equalit y are also hold for the new canon- ical con v o lution op erato r asso ciated with the LCT . Finally , we inv estig ate the sufficient and necessary conditions for solving a class of conv olutio n equatio ns asso ciated with the LCT. Keywor ds. Con v olution op erator; Con v o lution theorem; Linear canonica l transform ; Y oung’s in equalit y; Conv o lution equations 1 In tro duction The linea r canonica l transform (LCT) [3, 4, 9, 24] is a class of lin ear int egral transform c haracterized with parameter A = ( a, b, c, d ). It is well k n o wn that F ourier transform, F r esnel transform, fractional F ourier trans f orm [19], and scaling op erations, are all sp ecial cases of the LCT by c ho osing sp ecific parameters of A . Th erefore, the LCT has recently dra wn m uc h atten tion as a p o w e rful mathematical to ol in the fields of signal pro cessing, comm unications and optics [10, 12]. So far, many classica l results in the F ourier transform domain ha v e b een extended to the LCT domain, for in s tance, sampling th eorem [8, 13, 20, 21, 26, 27], uncertain t y principle [7, 14, 18, 28, 29], con v olution theorem [5, 11, 15, 22, 25], etc. In this pap er, w e revisit the con v olution theorem for the LCT. As w e all kno wn, the classical con v olution theorem for the F ourier transform states that the F ou r ier transform of the conv o lution of t w o functions is equal to the p oin t w ise pro d uct of the F ourier transforms. Th at is to sa y , in the F ourier transform domain, the classical con v olution transform is expressed as: F ( f ∗ g )( u ) = F f ( u ) F g ( u ) , (1) where the con v olution op erator ∗ is defin ed by f ∗ g ( t ) = Z + ∞ −∞ f ( τ ) g ( t − τ )d τ . (2) 1 Unfortunately , this claim is not tru e f or the LCT. Th erefore, by defin ing d ifferen t forms of con v olution op er ators (called canonical conv olution op erators in order to distinguish fr om the aforement ioned conv olutio n op erator asso ciated with the F ourier tr ansform), a v a riet y of con v olution theorems for the LCT ha v e b een deriv ed, see for example, Pei and Ding [11], Deng et al. [5], W ei et al. [22, 23], Shi et al. [15, 16]. P ei and Ding [11] in tro duced a canonical con v olution op erator O A conv , which is d en oted as O A conv ( f ( t ) , g ( t )) = L A − 1  L A f ( u ) L A g ( u )  ( t ) = r 1 j 8 π 3 b 3 Z R 3 e j a 2 b ( v 2 + u 2 + τ 2 − t 2 ) − j u b ( t − τ − v ) × f ( v ) g ( τ )d v d u d τ . (3) Then, the con v olution theorem asso ciated with the LCT can b e expressed as follo ws L A { O A conv ( f ( t ) , g ( t )) } ( u ) = L A f ( u ) L A g ( u ) , (4) whic h is similar to the traditional con v olution theorem in the F our ier transform d omain. Ho w ev er, we can see from (3) that it is difficult to redu ce O A conv ( f ( t ) , g ( t )) into a single in tegral form as in the traditional con v olution op erator formula (2). Deng et al. [5] p rop osed another canonical conv ol ution op erator Θ, which is defined b y ( f Θ g )( t ) = r 1 j 2 π b e − j a 2 b t 2   e j a 2 b t 2 f ( t )  ∗  e j a 2 b t 2 g ( t )   = Z + ∞ −∞ f ( τ ) g ( t − τ ) e − j a b τ ( t − τ ) d τ . (5) Th us, the con v olution theorem in the LCT domain can b e represented as L A ( f Θ g )( u ) = L A f ( u ) L A g ( u ) e − j d 2 b u 2 . (6) Later, W ei et al. [22] indep endently in v estigated the conv o lution th eorem (6) and obtained some extended results. Moreo v er, W ei et al. [23] p rop osed a new canonical con v olution op erator A Θ as follo ws  f A Θ g  ( t ) = Z + ∞ −∞ f ( τ ) g ( tθ τ )d τ . (7) Here, g ( tθ τ ) is the τ -generalize d translation of g ( t ), wh ic h is defined b y g ( tθ τ ) = r 1 j 2 πb r 1 − j b e − j a 2 b ( t 2 − τ 2 ) × 1 √ 2 π Z + ∞ −∞ L A g ( u ) e j 1 b ( t − τ ) u d u. (8) Hence, the con v olution theorem asso ciated with the LC T b ecomes L A  f A Θ g  ( u ) = L A f ( u ) L A g ( u ) . (9) 2 The form (7) is also quite simple with resp ect to that of F ourier transf orm . F u rthermore, Shi et al. [16] introd uced a n ew canonical con v olution structure for the LCT, and the canonical con v olution op erato r Θ M is denoted as ( f Θ M g )( t ) = Z + ∞ −∞ f ( τ ) g ( t − τ ) e − j a b τ ( t − τ 2 ) d τ . (10) Therefore, the con v olution theorem has the follo wing form L A ( f Θ M g )( u ) = √ 2 π L A f ( u ) F g  u b  . (11) It is sho wn in [16] that the new canonical con v olutio n op erator is quite useful for signal pro cessing. Later, Sh i et al. [15] prop osed another canonical con v olution op erato r in the follo wing ( f Ξ A 1 ,A 2 ,A 3 g )( t ) = Z + ∞ −∞ (T A 1 τ f )( t ) g ( τ ) ρ a 1 ,a 2 ,a 3 ( t, τ )d τ . (12) Here, (T A 1 τ f )( t ) = f ( t − τ ) e − j a 1 b 1 τ ( t − τ 2 ) , (13) and ρ a 1 ,a 2 ,a 3 ( t, τ ) = e j a 2 b 2 τ 2 + j  a 1 2 b 1 − a 3 2 b 3  t 2 . Then, the con v olution theorem for the LCT has the form L A 3 ( f Ξ A 1 ,A 2 ,A 3 g )( u ) = ǫ d 1 ,d 2 ,d 3 ( u ) L A 1  b 1 b 3 u  L A 2  b 2 b 3 u  , (14) where ǫ d 1 ,d 2 ,d 3 ( u ) = r j 2 πb 1 b 2 b 3 e j u 2  d 3 2 b 3 − d 1 b 2 1 2 b 1 b 2 3 − d 2 b 2 2 2 b 2 b 2 3  . It follo ws from [15] that the classical con v olution theorem for the F ourier transform, the generalized con v olution theorem for the fractio nal F ourier transform, and some existing canonical co nv o lution theorems asso ciat ed with th e LCT can b e regarded as the sp ecial cases for (14). The canonical con v ol ution op erators for the LCT in trodu ced in the six pap ers men- tioned ab ov e are very in teresting, and can b e ap p lied to solving many theoretical or prac- tical problems, since they can b e considered as some extensions of classica l con v olutio n op erator for the F ou r ier transf orm. In this p ap er, our goal is to int ro d u ce a n ew canonical con v olution op erator f or th e LCT, and then derive a generalize d ve rsion of the classical con v olution theorem, and Y oung’s inequalit y asso cia ted w ith the F ourier transform. F ur- thermore, we discuss th e solv ability of a class of con v ol ution equations asso ciated w ith the new canonical con v olution op erator. In Sec 3, w e v erify that our new defined canoni- cal con v olution op erator can b e p erformed into t w o different w a ys for implement in filter 3 design. T his fact may hav e some adv an tag es ov e r others in filter d esign, for example, com- pared with the canonical con v olution op erators in tro duced in references [5 , 15, 16, 22, 23], whic h only ha v e one wa y of conv o luting, resp ectiv ely . In fact, by consid ering the compu - tational complexit y or input conditions, w e can ha v e tw o options for c ho osing fi ltering, since in some cases, the first one ma y b e p erform b etter than the second one, or vice versa. Therefore, when app lied to solving some sp ecific problems, our canonical con v olution in- tro duced in this pap er is muc h more fl exible than th e existing ones for the LCT men tioned in [5, 15, 16, 22, 23]. The rest o f p ap er is organized as follo ws. In Sec 2 , we briefly recall the d efinition of the LCT. In Sec 3, w e in tro duce a new canonical con v olutio n op erator, and pro v e that it satisfies the generalized conv olutio n theorem for the LC T. In Sec 4, w e present t wo applications for the new canonical con v olution op erator. First, we deriv e a generalize d Y oung’s inequalit y for the new canonical conv o lution operator asso ciated with the LCT . Second, w e giv e some sufficien t a nd necessary co nditions for the solv abilit y of a class of con v olution equations asso cia ted with our new defined canonical con v olution op erator. Finally , w e conclude the pap er. 2 The Linear Canonical T ransform Definition 2.1. The LCT of a signal f ( t ) ∈ L 1 ( R ) is define d by [17]: L A f ( u ) := L A { f ( t ) } ( u ) =    r 1 j 2 π b Z + ∞ −∞ f ( t ) e j a 2 b t 2 − j 1 b ut + j d 2 b u 2 d t, b 6 = 0 , √ de j cd 2 u 2 f ( du ) , b = 0 . (15) wher e A = ( a, b, c, d ) , and p ar ameter s a, b, c, d ∈ R satisfy ad − bc = 1 . F or b = 0, the LCT b ecome s a Chirp m ultiplicatio n op erator. Hence, without loss of generalit y , w e a ssume that b 6 = 0 in the rest of the pap er. As aforemen tioned, the LCT includes many linear inte gral trans f orms as sp ecial cases. F or instance, let A = (0 , 1 , − 1 , 0) , then the LCT (15) r ed uces to the F our ier transform; let A = (cos α, sin α, − sin α, cos α ) , then the LCT (15) b ecomes to the fractional F ourier transform. 3 A New Generalized Con v o lution Theorem for the LCT In this section, we fi rst in tro duce a new canonical con v olution op erator whic h is quite differen t from the existing ones. It is sh o wn that our new canonical conv o lution op erator is m uc h more fl exible and useful in certain cases. Then , we stud y the corresp ond ing gen- eralized conv olution th eorem asso cia ted with the LCT. Finally , we giv e s everal prop erties that the new canonical con v olution op erator satisfies. First, we in tro du ce a new notion of canonical con v olutio n op erator, w hic h is related to the LCT parameter A . Our new definition is a generalized v ersion of [2, Definition 1]. Definition 3.1. Given two fu nc tions f , g ∈ L 1 ( R ) , the c anonic a l c onvolutio n op er ator ⊗ A is denote d as ( f ⊗ A g )( t ) = r 1 j 2 πb Z + ∞ −∞ f ( u ) g ( t − u + b ) 4  Delay Convolution ሺሻ ሺሻ ሺ ൅ ሻ  ୨ ௔ ଶ௕ ௦ మ  ୨ ௔ ଶ௕ ௦ మ ା௝௔௦ ඨ ͳ ݆ʹߨܾ  ି୨ ௔ ଶ௕ ௧ మ ሺሻ Figure 1: One expression of the canonical con v olution op erator ⊗ A × e j a b u 2 − j a b ut + j at − j au d u, (16) wher e A = ( a, b, c, d ) is define d the same as the LCT p ar ameter. The new canonical con v olution exp r ession (16) can b e r ewritten in to tw o differen t forms, according to the classical con v olution op erator ∗ . Firs t, it can b e r epresen ted as h ( t ) :=( f ⊗ A g )( t ) =( e j a 2 b s 2 · f ( s )) ∗ ( e j as · e j a 2 b s 2 · g ( s + b ))( t ) × r 1 j 2 πb e − j a 2 b t 2 . (17) Second, it can also b e describ ed as h ( t ) :=( f ⊗ A g )( t ) =( e − j as · e − j a 2 b s 2 · f ( s )) ∗ ( e j a 2 b s 2 · g ( s + b ))( t ) × r 1 j 2 πb e j a 2 b t 2 + j at . (18) Thanks to (17) and (18), we giv e t w o realizations for the new canonical conv o lution op er- ator ⊗ A in Fig. 1 , and Fig. 2, resp ectiv ely . Compared to the canonical conv o lution op erator defined in (3), our n ew canonical con v olution is a sin gle in tegral, w hic h is m uc h simpler than the trip le in tegral men tioned in (3). F urthermore, th e definition of canonical con v olution op erato r in (12) is s o complicated that it is n ot quite u seful in filter design. Although the f orm of the new defined canonical con v olution op erato r (16) is similar to those of (5), (7) and (10), it follo ws from (17) and (18) th at th e new canonical con v olution op erator ⊗ A has tw o realizations in fi ler design. Therefore, in certain cases, our new canonical con v olution op erator is muc h more flexible and useful than those in [5, 11, 15, 16, 22, 23]. Based on the new canonical con v olution op erato r ⊗ A , we derive the generalized con- v olution theorem asso ciated with th e LCT as follo ws. 5 Delay Convolution ሺሻ ሺሻ ሺ ൅ ሻ  ି୨ ௔ ଶ௕ ௦ మ ି௝௔௦  ୨ ௔ ଶ௕ ௦ మ ඨ ͳ ݆ʹߨܾ  ୨ ௔ ଶ௕ ௧ మ ା௝௔௧ ሺሻ Figure 2: Alternativ e representat ion of the canonical conv ol ution op erator ⊗ A Theorem 3.2. L et f , g ∈ L 1 ( R ) , Φ( u ) := e j u − j d 2 b u 2 − j ab 2 . Then, we have k f ⊗ A g k 1 ≤ s 1 2 π | b | k f k 1 k g k 1 , (19) and L A ( f ⊗ A g )( u ) = Φ( u ) L A f ( u ) L A g ( u ) . (20) Pr o of. First, we pr o v e (19). Let s = t − u + b . By the definition of canonical conv olution op erator ⊗ A (16), w e ha v e k f ⊗ A g k 1 = Z + ∞ −∞ | ( f ⊗ A g )( t ) | d t ≤ s 1 2 π | b | Z + ∞ −∞ Z + ∞ −∞ | f ( u ) g ( t − u + b ) | d u d t = s 1 2 π | b | Z + ∞ −∞ | f ( u ) | d u Z + ∞ −∞ | g ( s ) | d s = s 1 2 π | b | k f k 1 k g k 1 . (21) Next, we prov e (20). By using the defi nition of the LCT, m aking c hange of v aria ble v = s − t + b , and then utilizing the d efinition of canonical con v olution op erator (16 ), w e ha v e Φ( u ) L A f ( u ) L A g ( u ) = e j u − j d 2 b u 2 − j ab 2 1 j 2 π b Z + ∞ −∞ f ( t ) e j a 2 b t 2 − j 1 b ut + j d 2 b u 2 d t × Z + ∞ −∞ g ( v ) e j a 2 b v 2 − j 1 b uv + j d 2 b u 2 d v 6 = e j u − j d 2 b u 2 − j ab 2 1 j 2 π b Z + ∞ −∞ Z + ∞ −∞ f ( t ) g ( v ) × e j a 2 b ( t 2 + v 2 ) − j 1 b u ( t + v )+ j d b u 2 d t d v = 1 j 2 π b e − j ab 2 Z + ∞ −∞ Z + ∞ −∞ f ( t ) g ( v ) × e j a 2 b ( t 2 + v 2 ) − j 1 b u ( t + v − b )+ j d 2 b u 2 d t d v = 1 j 2 π b e − j ab 2 Z + ∞ −∞ Z + ∞ −∞ f ( t ) g ( s − t + b ) × e j a 2 b [ t 2 +( s − t + b ) 2 ] − j 1 b us + j d 2 b u 2 d s d t = 1 j 2 π b e − j ab 2 Z + ∞ −∞ Z + ∞ −∞ f ( t ) g ( s − t + b ) × e j a 2 b [2 t 2 − 2 st + s 2 +2 bs − 2 bt + b 2 ] − j 1 b us + j d 2 b u 2 d s d t = r 1 j 2 π b Z + ∞ −∞ e j a 2 b s 2 − j 1 b us + j d 2 b u 2 ×  r 1 j 2 π b Z + ∞ −∞ f ( t ) g ( s − t + b ) e j a b t 2 − j a b st + j as − j at d t  d s = r 1 j 2 π b Z + ∞ −∞ e j a 2 b s 2 − j 1 b us + j d 2 b u 2 ( f ⊗ A g )( s )d s = L A ( f ⊗ A g )( u ) . (22) This completes the p ro of. After simple computation, it follo ws from Th eorem 3.2 that the canonical con v olution op erator ⊗ A satisfies thr ee prop erties: Commutati v e prop er ty , associativ e prop erty , a nd distributive pr op erty . More clearly , the foll o wing th ree equalities hold for any f , g , h ∈ L 1 ( R ): (1) Commutat ivit y: f ⊗ A g = g ⊗ A f . (2) Asso ciativit y: ( f ⊗ A g ) ⊗ A h = f ⊗ A ( g ⊗ A h ). (3) Distributivit y: f ⊗ A ( g + h ) = f ⊗ A g + f ⊗ A h . 4 Tw o Applications for the New Canonical Con v olution Op- erator 4.1 Generalized Y oung’s I nequalit y In this subsection, we inv esti gate the generalized Y oung’s inequality for th e new canonical con v olution op erato r ⊗ A . First, let us recall the classical Y oung’s in equalit y as follo ws. Prop osition 4.1 ([6]) . L e t f ∈ L p ( R ) , g ∈ L q ( R ) , 1 p + 1 q = 1 + 1 r , 1 r + 1 r ′ = 1 . Then, k f ∗ g k r ≤ A p A q A r ′ k f k p k g k q , (23) 7 wher e A p =  p 1 /p p ′ 1 /p ′  1 / 2 , (24) and 1 /p + 1 /p ′ = 1 . Next, w e sh o w that our n ew canonical conv olution op erator ⊗ A also s atisfies the Y oung’s in equalit y . Theorem 4.2. L et f ∈ L p ( R ) , g ∈ L q ( R ) , 1 p + 1 q = 1 + 1 r , 1 r + 1 r ′ = 1 . Then, k f ⊗ A g k r ≤ s 1 2 π | b | A p A q A r ′ k f k p k g k q , (25) wher e A p is define d the same as in (24). Pr o of. By (17), we obtain k f ⊗ A g k r =  Z + ∞ −∞    ( e j a 2 b s 2 · f ( s )) ∗ ( e j as · e j a 2 b s 2 · g ( s + b ))( t ) × r 1 j 2 π b e − j a 2 b t 2    r d t  1 r = s 1 2 π | b |  Z + ∞ −∞     ( e j a 2 b s 2 f ( s )) ∗ ( e j as + j a 2 b s 2 g ( s + b ))( t )     r d t  1 r = s 1 2 π | b |    ( e j a 2 b ( · ) 2 f ( · )) ∗ ( e j a ( · )+ j a 2 b ( · ) 2 g ( · + b ))    r , s 1 2 π | b | k ˜ f ∗ ˜ g k r , (26) where ˜ f ( · ) , e j a 2 b ( · ) 2 f ( · ), ˜ g ( · ) , e j a ( · )+ j a 2 b ( · ) 2 g ( · + b ). Note that ˜ f ∈ L p ( R ) , ˜ g ∈ L q ( R ). Applying the classical Y oung’s inequalit y (23) for the functions ˜ f and ˜ g , we ha v e k ˜ f ∗ ˜ g k r ≤ A p A q A r ′ k ˜ f k p k ˜ g k q , (27) Note that k ˜ f k p = k f k p , k ˜ g k q = k g k q . Sub stituting (27) into (26), we get k f ⊗ A g k r = s 1 2 π | b | k ˜ f ∗ ˜ g k r ≤ s 1 2 π | b | A p A q A r ′ k ˜ f k p k ˜ g k q = s 1 2 π | b | A p A q A r ′ k f k p k g k q , (28) whic h completes the pro of. 8 4.2 Solv abilit y for One Class of Con v olution Equations In this sub section, we mainly discuss the solution for a class of con v olution equations asso ciated with the canonical conv o lution op erator ⊗ A . Assu me that λ ∈ C , a nd f , g ∈ L 1 ( R ) are given, φ is unknown, consid er the follo wing canonical con v olution equation: λφ ( t ) + ( g ⊗ A φ )( t ) = f ( t ) . (29) In the sequel, we w ill determine the v a lue of φ . Before p resen ting our main result, we giv e a lemma, whic h is v ery imp ortant for proving our theorem. Lemma 4.3. L et Λ( u ) := λ + L A g ( u )Φ( u ) , then the fol lowing two statements hold: ( a ) If λ 6 = 0 , then ther e exists a c o nstant C, such that Λ( u ) 6 = 0 for every | u | > C . ( b ) If for al l u ∈ R , Λ( u ) 6 = 0 , then 1 Λ( u ) is c ontinuous and b ounde d on R . The p ro of of Lemma 4.3 is similar to those of [1, Prop ositio n 7] and [2, Pr op osition 1], hence, w e omit the pro of. Theorem 4.4. L et Λ( u ) 6 = 0 f or al l u ∈ R . Sup p o se that one of the f ol low ing two c o nditions holds: (1) λ 6 = 0 , and L A f ∈ L 1 ( R ) ; (2) λ = 0 , and L A f L A g ∈ L 1 ( R ) . Then e quation (29) has a solution in L 1 ( R ) if and only if L A − 1  L A f Λ  ∈ L 1 ( R ) . F urther- mor e, the solution has the form of φ = L A − 1  L A f Λ  . Pr o of. W e only consider the case w hen th e condition (1) is satisfied. Since Φ ( u ) = e j u − j d 2 b u 2 − j ab 2 , | Φ( x ) | = 1, we kno w that 1 Φ is con tin uous and b oun ded on R . Hence, L A f L A g ∈ L 1 ( R ) if and only if L A f Φ L A g ∈ L 1 ( R ). Therefore, the case (2) b ecomes to the case (1). Necessit y: S u pp ose that equation (29 ) has a solution φ ∈ L 1 ( R ). Multiplying the op erator L A to the b oth sides of the equation (29), then w e ha v e λ L A φ ( u ) + L A ( g ⊗ A φ )( u ) = L A f ( u ) . (30) By using (20), w e obtain λ L A φ ( u ) + Φ( u ) L A g ( u ) L A φ ( u ) = L A f ( u ) , (31) i.e., ( λ + Φ( u ) L A g ( u )) L A φ ( u ) = L A f ( u ) . (32) Since λ 6 = 0, then Λ( u ) = λ + Φ ( u ) L A g ( u ) 6 = 0 for all u ∈ R . Th erefore, the equation (32) b ecomes L A φ ( u ) = L A f ( u ) Λ( u ) . (33) 9 By Lemma 4.3, we know that 1 Λ( u ) is con tin uous and b ounded on R . Sin ce L A f ∈ L 1 ( R ), w e ha v e L A f ( u ) Λ( u ) ∈ L 1 ( R ). T aking in v erse LCT transform to the b oth sides of equation (33), we obtain the solution φ ( t ) = L A − 1  L A f ( u ) Λ( u )  ( t ) . Since φ ∈ L 1 ( R ), then L A − 1  L A f Λ  ∈ L 1 ( R ) . Sufficiency: Let φ ( t ) := L A − 1  L A f ( u ) Λ( u )  ( t ) . Then, w e ha v e φ ∈ L 1 ( R ). Applying the LCT to φ , w e get L A φ ( u ) = L A f ( u ) Λ( u ) . That is to sa y , ( λ + Φ( u ) L A g ( u )) L A φ ( u ) = L A f ( u ) . By using (20) again, w e obtain L A { λφ ( t ) + ( g ⊗ A φ )( t ) } ( u ) = L A f ( u ) . Due to the u niqueness of LCT op erator L A , φ satisfies the equation (29) for almost ev ery t ∈ R , which m eans that equation (29) has a s olution. This completes the pro of. Corollary 4.5. L et A = (cos α, sin α, − sin α, cos α ) , then The or em 4.4 r e duc es to the The or em 3 mentione d in [2]. Remark 4.6. Similar to the Definition 3.1, we c an define another c anonic al c o nvolution op er ato r ⊙ A by  f ⊙ A g  ( t ) = r 1 j 2 πb Z + ∞ −∞ f ( u ) g ( t − u − b ) × e j a b u 2 − j a b ut − j at + j au d u. (34) Then, the new c ano nic al c onvo lution op er at or ⊙ A also has thr e e pr op erties: Commutative pr op ert y, asso ciative pr o p e rty, and distributive pr o p er ty. In addition, the statements in The or em 3.2, The o r em 4.2, and The or em 4.4 also ho ld for op e r ator ⊙ A with some minor adjustments. Due to the similarity, we omit the pr o of of this claim. 5 Conclusion In this pap er, we first define a new canonica l con v olutio n op erator, whic h is muc h more flexible and simp le than the existing ones. Then , we sho w that it satisfies the generalized con v olution theorem and Y oung’s inequalit y . 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