Int. J. Anal. Appl. (2022), 20:33 Received: May 16, 2022. 2010 Mathematics Subject Classification. 42C40, 65R10, 44A35. Key words and phrases. continuous wavelet transform, q-Bessel Fourier transform, q-Bessel operator. https://doi.org/10.28924/2291-8639-20-2022-33 © 2022 the author(s) ISSN: 2291-8639 1 The Continuous Wavelet Transform for a q-Bessel Type Operator C.P. Pandey1,*, Jyoti Saikia2 Department of Mathematics, North Eastern Regional Institute of Science and Technology, Nirjuli, 791109, Arunachal Pradesh, India *Corresponding author: drcppandey@gmail.com ABSTRACT. In this paper, we consider a differential operator  on )0, - By accomplishing harmonic analysis tools with respect to the operator  we study some definitions and properties of q-Bessel continuous wavelet transform. We also explore generalized q-Bessel Fourier transform and convolution product on  )0, associated with the operator  and finally a new continuous wavelet transform associated with q-Bessel operator is constructed and investigated. 1. Introduction For a function ( )2f L R , the wavelet transform with respect to the wavelet ( )2L R  is defined by ( ) ( ) ( ) 2 12 1 , 2 1 ( ) , , , 0W f f t t dt R          − =   (1.1) where, 2 1 1/ 2 2 , 1 1 ( ) . t t        −  − =     (1.2) https://doi.org/10.28924/2291-8639-20-2022-33 2 Int. J. Anal. Appl. (2022), 20:33 Translation 2  is defined by 2 2 2 ( ) ( ),t t R      = −  and dilation 1 D  is defined by 1 1/ 2 1 1 1 ( ) , 0 t D t       −   =     . We can write ( ) ( ) 2 1 2 1, t D t       = . (1.3) From above equations, we can say that wavelet transform of the function f on R is an integral transform and the dilated translate of  is the kernel. We can also express wavelet transform as the convolution: ( )( ) ( )( ) 12 1 , 2 , * o W f f g     = , (1.4) Where, ( ) ( ).g t t= − Since there is a special type of convolution for every integral transform, therefore one can define wavelet transform with respect to a integral transform using associated convolution. The concept of wavelet is a collection of function derived from a single function called mother wavelet, after that by applying the two operators known as translation and dilation we get a new type of continuous wavelet transform. Here presently, we introduce a q-Bessel operator [1] and [2]. ( ) ( ) ( ) ( )( )1 2 2, 2 1 ( ) 1 . v v q v f t f q t q f t q f qt t −  = − + + (1.5) The above q-Bessel operator associated with q-Bessel function by the eigenvalue equation. ( ) ( )2 2 2, , , .q v v vj x q j x q = − Unlike the elementary functions such as trigonometric, exponential etc the Bessel wavelets are related to special functions and Jachkson introduced the concept of q-analysis at the beginning of the twentieth century. We have arranged this paper as follows: In section 2, we will review briefly the basics of q-Bessel Fourier transform, here we recall notations, some definitions of q-Bessel Fourier and Inverse Fourier transform and the preposition associated with other operators and convolution 3 Int. J. Anal. Appl. (2022), 20:33 product. In section 3, some results of harmonic analysis with respect to q-Bessel operator for the generalized q-Bessel transform is collected and the definition and properties of convolution product is also discussed. To extend the classical theory of wavelets to the differential operator ,q   is the actual aim of this work. We define a generalized wavelet, which satisfy the below admissibility condition ( )( ) , 2 , , 0 0 . q q q g d C F g        =   (1.6) Where ,q F  denotes the generalized q-Bessel Fourier transform related to operator given by ( )( ) ( ) ( ) , 2 2 1 , 0 , q q q F g c g t j t q t d t        +  =  ( ), , .q p qg L  +   With ( ) ( ) 2 2 2 , 2 2 ;1 , 1 ; q q q c q q q   +   = − and ( );j x q being the normalized Bessel function of index  . Starting with a single generalized wavelet g, a family of generalized wavelets is constructed by putting ( ) ( )( )   1 2 , , , , 0 , a b q b a q q g x a T g x a b  + + =      where ( ) 2 2 2 1 a n x g x g a a + +   =     and ,q b T  is generalized translation operators related to the differential operator ,q   . The continuous generalized q-Bessel wavelet transform of a function  ( ),2, 0q qf L  +   at the scale q a +  and the position  0qb +   is defined by ( )( ) ( ) ( ) 2 1 , , , , 0 , . q g q qa b f a b c f x g x d x       + =  (1.7) In section 4, we develop a relationship between the generalized wavelet transforms and q-Bessel continuous wavelet transforms. Such a relationship helps us to build certain formulas for the generalized q-Bessel continuous wavelet transform (CWT). 4 Int. J. Anal. Appl. (2022), 20:33 In Section 5, we study the intertwining operator q  to establish the continuous generalized q- Bessel wavelet transform in form of classical one. As a result, we got a new inversion formulas for dual operator t q  of q  . 2. Preliminaries In the present section we recapitulate some facts about harmonic analysis related to the q- Bessel operator. We cite here, as briefly as possible, only those properties actually required for the discussion. Throughout this section assume 1/ 2  − . Let the space , ,q p L  , 1 p   denote the sets of real functions on q + for which ( ) 1/ 2 1 , , 0 , P P qq p f f x x d x    +   =       and ( ) , , . q q x f Sup f x  +   =   The q-Bessel Fourier transform , ,,q n F  in [3] is defined for ,1,q f L   by ( )( ) ( ) ( )2 2 1, , 0 , , , q q q q F f c f t j t q t d t t        + + =   (2.1) where j  is normalized q-Bessel function. ( ) ( ) ( ) ( ) ( ) 1 2 2 2 2 2 2 2 0 , 1 . ; , n n n n n n q j x q x q q q q   + + = = − (2.2) Theorem 2.1 (i) The q-Bessel Fourier transform , ,2, ,2, : q q q F L L    → defines an isomorphism and for all functions ,2,q f L   , ( ) ( )2, , ,2,,2,, .q q qqF f f F f f  = = (2.3) (ii) If f , ( ),qF f ,1,qL  then ( ) ( )( ) ( ) ( )2, , 0 , , q n q f x F f j x q d        =  (2.4) for almost all , q x +   where 5 Int. J. Anal. Appl. (2022), 20:33 ( ) ( ) ( )2 2 1 1 1 q q q q d d         − + + = + (2.5) (iii) For all ,1, ,2,q q f L L     we have ( ) ( ) 2 22 1 2 1 , 0 0 . q q q F f d f x x d x        + + =  (iv) The inverse transform is given by ( )( ) ( ) ( ) ( )1 2, , 0 , , q n q F g x g j x q d        − =  The q-Bessel translation operators , , 0, q x x    is defined by ( )( ) ( ) ( ) 2 1, , 0 , , , q x q q f y f z D x y z z d z      + =  (2.6) where ( ) ( ) ( ) ( )2 2 2 2 2 1, , 0 , , , , , q q q D x y z c j xs q j ys q j zs q s d s        +   =      (2.7) The convolution product of q-Bessel for two functions ,f g is defined as ( ) ( ) ( ) 2 1, , 0 , 0. q q q x q f g x c f y g y y d y x      +  =   (2.8) Theorem 2.2 (i) Let 1 p   and , ,q p f L   . Then 0x  , , , ,q x q pL     and , , ,, , . q x q pq p f f     (ii) For , ,q p f L   , 1 p   , we have ( )( ) ( ) ( )( )2, ,, , , ,,, .q n q x q nF f j x q F f       = (iii) Let , [1, )p r   such that 1 1 1 p r + = . If , ,q p f L   and , ,q p g L   , then for every 0x  we have ( ) ( ) ( ) ( )2 1 2 1, , 0 0 q x q q x q f y g y y d y f y g y y d y         + + =  (iv) For , , [1, )p r s   such that 1 1 1 1 p r s + − = . If , ,q p f L   and , ,q p g L   then 6 Int. J. Anal. Appl. (2022), 20:33 , , , ,, , q q p q rq s f g f g     (v) For , ,q p f L   and , ,q p g L   we have ( ) ( ) ( ), ,, , ,, , ,, .q n q q n q nF f g F f F g   = Definition 2.1 A function ,2,q g L   is a q-Bessel wavelet of order  , if it satisfies the admissibility condition. ( )( ) 2 , , ,, 0 0 . q g q n d C F g        =   (2.9) Definition 2.2 Let  ( ),2, 0q qg L  +   be a q-Bessel wavelet of order  . Then continuous q-Bessel wavelet transform is defined as follows ( )( ) ( ) ( )   2 1 , , , 0 , , , 0 , q g q q q qa b S f a b c f x g x d x a b      + + + =      (2.10) where ( ) ( ) 1 2 ,, , , q b a qa b g a g a b    + =   (2.11) 2 2 1 a x g g a a +   =     (2.12) The q-Bessel continuous wavelet transform has been investigated in detail in [4] from which we see the following basis properties. Theorem 2.3 Let be  ( ),2, 0q qg L  +   be a q-Bessel wavelet. Then (i) For all  ( ),2, 0q qf L  +   , the Plancherel formula we have ( ) ( )( ) 22 2 1 2 1 , 2 ,0 0 0 1 , q q q g q g d a f x x d x S f a b b d b C a        + + =  . (ii) For all  ( ),2, 0q qf L  +   , we have ( ) ( )( ) ( ) , 2 1 , , 2 , 0 0 , , q q q g q qa b g c d a f x S f a b g b d b x C a        + +   =        . 3. Harmonic Analysis Associated with  and Generalized Fourier Transform Let M be the map defined by ( ) ( )2 .nMf x x f x= 7 Int. J. Anal. Appl. (2022), 20:33 Let , ,q p L  , 1 p   be the class of measurable functions f on [0, ) for which 1 , , , , , 2 . q p n q p n f M f   − + =   For  and x , put ( ) ( )2 2 2, 2, , . n n n x q x j x q     + = (3.1) where 2n j + is the normalized Bessel function with index 2n + is given by equation (2.1). From [4] see the following properties. Theorem 3.1 (i) ,n  possess the Laplace type integral representation ( ) ( ) ( ) ( ) ( ) 1 2 2 2 2 2 , 0 , 1 : : cos : n n q q q C q x F t q xt q d t     = +  (3.2) when 1q − → and 1 2  −  where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 12 2 2 2 2 2 1 2 0 2 1 ; 1 : , : , cos : 1 . 1 1 ( ; ); 2 2 n nq n n n n n q q x q q q C q F t q x q q x q qx q q       − + =   + − = = = −       +         (ii) ( )2, ,n q  satisfies the differential equation ( ) ( )2 2 2, , , ,, ,q n n nq q       = − . (3.3) (iii) For all  and x ( ) Im2 2, , xn n q x e      . (3.4) Definition 3.1 The generalized q-Bessel Fourier transform is defined for a function ,1, ,q n f L   is defined by ( )( ) ( ) ( )2 2 1, , 2 , 0 , q q n n q F f c f x x q x d x        +  + =  (3.5) By (3.1) and (3.5) we observe that 1 , , 2 , q q n F F M  −  + = (3.6) where , 2q n F + is the Fourier-Bessel transform of order 2n + . (ii) If ,1, ,q n f L   then ( ) ( ), 0 [0, )qF f C   and 8 Int. J. Anal. Appl. (2022), 20:33 , , , 2 ,1, ,, , , ( ) q n q n q nq n F f B f    +  where , 2q n B + is given in [3]. Theorem 3.2 Let ,1, ,q n f L   such that ( ), ,1, 2q q nF f L  + . Then for almost all 0x  , ( ) ( )( ) ( )2 2 1, 2 , , 0 , . q n q n q f x c F f x q d          + +  =  Proof. By (3.1), (3.6) and theorem 2.1(ii) we have ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 1 2 1 2 1 , 2 , , , 2 , 2 2 0 0 2 1 2 1 , 2 , 2 2 0 2 1 , , n q n q n q q n q n n q n q n q n n q n c F f x q d x c F M f j x d x M c F f j x d x M f x f x                           + − + +  + + +  − + + + + − = = = =    for all 0x  . Theorem 3.3 (i) For every ,1, , , , ,q n q p n f L L     space where 2p  we have the Plancherel formula ( )( ) ( )( ) ( ) 22 2 1 , 2 0 0 . q q q n f t t d t F f d       +  + =  (ii) The inverse of this transform is given by ( )( ) ( ) ( ) ( )1 2, , 2 0 , q n q n F g x g x q d         −  + =  . Proof. (i) Let ,1, , , , ,q n q p n f L L     . By (3.6) and theorem 2.1 (iii) we have ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( ) 22 1 , 2 , 2 2 0 0 2 1 2 4 1 0 2 2 1 0 , q q n q n q n n q q F f d F M f d M f x x d x f x x d x             −  + + +  − + +  + = = =     The proof of (ii) is standard. 4. Generalized Convolution Product Definition 4.1 The generalized translation operator , ,q x n T  is define by the relation 9 Int. J. Anal. Appl. (2022), 20:33 2 2 1 , , , n n q x n q x T x M M    + − = (4.1) where 2 , n q x   + are the Bessel translation operators of order 2n + . Definition 4.2 Define the generalized convolution product of two functions f and g on [0, ) by ( ) ( ) ( ) 2 1, 2 , , 0 # q q n q x n q f g x c T f y g y y d y     + + =  (4.2) where , 2q n c + is given by (1.6). From by (4.1) we have ( ) ( )1 1, 2# ,q q nf g M M f M g − − +  =    (4.3) where , 2q n+  is the Bessel convolution. Theorem 4.1 (i) Let f be in ,1, , , 1 . q n L p     Then 2 , , ,1, ,,1, , n q x n q nq n T x f    . (ii) For ,2, ,q n f L   , we have ( )( ) ( ) ( )( )2, , , , , ,,q q x n n q nF T f x q F f         = . (iii) If ,1, ,q n f L   and ,1, ,q n g L   then ( ) ( ) ( ) ( )2 1 2 1, , , , 0 0 . q x n q q x n q T f y g y y d y f y T g y y d y       + + =  (iv) For ,1, , , q n f g L   then ,1, , # q q n f g L   and ,1, , ,1, ,,1, , # q q n q nq n f g f g    . (v) For ,1, ,q n f L   and ,1, ,q n g L   we have ( )( ) ( )( ) ( )( ), , ,#q q q qF f g F f F g    = . Proof. (i) By (4.1) and Theorem 2.2(i) we have 2 2 1 , , , ,1, , ,1, , 2 2 1 , ,1, 2 2 1 ,1, 2 2 ,1, , . n n q x n q x q n q n n n q x q n n q n n q n T f x M M f x M f x M f x f           + − + − + − + = =  = 10 Int. J. Anal. Appl. (2022), 20:33 (ii) By (3.1), (3.6), (4.1) and Theorem 2.2(ii) we have ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) 1 2 2 2 1 , , , , 2 , 2 2 1 2 1 , 2 , 2 2 1 1 2 , 2 2 1 , , 2 2 , , , , . n n n q q x n q n q x n n n q n q x n n n q n n q n n q F T f F M x M f x M F M f x M j F M f x q F M f x q F f                          − + −  + − + − + − − + + − +  = = = = = (iii) By (4.1) and Theorem 2.2(iii) we have ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 4 2 1 1 2 1 , , , 0 0 2 4 1 2 1 2 1 , 0 22 1 2 1 2 1 , 0 2 1 2 1 , , 0 2 1 , , 0 . n n n q x n q q x q n n n q x q nn n q x q n q x n q q x n q T f y g y y d y x y M f y M g y y d y x y M f y M g y y d y y M f y xy M g y y d y y M f y T g y y d y f y T g y y d y                  + + − − +  − + − +  − + − +  − +  + = = = = =       (iv) By (4.3) and Theorem 2.2(iv) we have ( )1 ,1, , ,1, 2 1 1 ,1, 2 ,1, 2 ,1, , ,1, , # # . q q q n q n q n q n q n q n f g M f g M f M g f g       − + − − + +   = (v) By (3.6), (4.3) and Theorem 2.2(v) we have ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 , , , 2 1 1 1 , 1 1 , , , , # * # . q q q q n q q q q q q F f g F M M f M g F M M M f M g F M f F M g F f F g         − −   + − − −  − −      =    =   = = This concludes the proof. 11 Int. J. Anal. Appl. (2022), 20:33 5. Transmutation Operators Definition 5.1 For a bounded function f on [0, ) , define the integral transform q  by ( ) ( ) ( ) ( ) ( ) 1 2 2 2 0 1 : : , n q q f x q C q x F t q f xt d t   = +  (5.1) where ( )2:C q and ( )2:F t q is given Theorem 3.1(i). Remark 5.1 (i) For n=0, q  reduces to q-Riemann Liouville integral transform of order  given by ( )( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 0, 1 : : , 0 0 , 0 . n q q q C q x F t q f xt d t if x R f x f x     +  =   =  (ii) It is checked that 2 ,q n q M R   + = (5.2) (iii) From Theorem 3.1(i) and (5.1) we have ( ) ( )( )( )2 2, , cos ,n qx q xt q x  = (5.3) Definition 5.2 Define the integral transform t q  for a differential function f on [0, ) by ( ) ( ) ( ) ( )2 2 21 : : qt q n qy d tx f y q C q F q f t t t      −   = +      Remark 5.2 (i) For n=0, t q  reduces to q-Weyl integral transform of order  given by ( )( ) ( ) ( ) ( ) 1 2 2 , 0 1 : : , 0 . q q y W f y q C q F q f t t d t y t       = +      (ii) It is seen that 1 2 , t q n q W M   − + = (5.4) Theorem 5.1 (i) If ( ), [0, ),qf L dx  then , ,q q nf L   and ,, , , .q qq nf f   (ii) If ,1, ,q n f L   then ( ),1 [0, ), t q q f L dx   and ,1, ,,1 . t q q nq f f    (iii) For any ( ),1 [0, ),qf L dx  and ,1, ,q ng L  we have the duality relation 12 Int. J. Anal. Appl. (2022), 20:33 ( ) ( ) ( ) ( )2 1 0 0 . t q q q q f x g x x d x f y g y d y      + =  (iv) For all ,1, ,q n f L   we have ( ) ( ), , , t q q C q F f F f  = (5.5) where ,q C F is the q-cosine Fourier transform given by ( )( ) ( ) ( )2, 0 cos ; , 0. q C q F f f x x q d x    =  (v) Let ,1, , , q n f g L   . Then ( )# * ,t t tq q q qf g f g  = where * is the convolution product defined by ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 2 1 2 0 1 1 x q q q f f x f y f y y d y     −  + +  =  +  , with x   is a q-generalized translation given in details in [5]. (vi) Let ,1, ,q n f L   and ( ), [0, ),qg L dx  . Then ( ) ( )* #tq q q qf g f g  = . (5.6) Proof. (i) By (5.1) and [5.2] we have , 2 , ,, , , , , , q q n q qq n q n f M R R f f     +    = =  (ii) By (5.1) and [5.4] we have 1 , ,1, ,,1 ,1, 2 t q q q nq q n f M R f    − +  = (iii) By (4.3), (5.2) we have ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) 2 1 1 2 4 1 2 , 0 0 1 2 , 0 0 . n q q n q q n q q t q q f x g x x d x R f x M g x x d x f y W M g y d y f y g y d y         + − + + +  − +  = = =     13 Int. J. Anal. Appl. (2022), 20:33 (iv) By (3.6), (5.4) we have ( ) ( ) ( ) ( ) 1 , , 2 , 1 , 2 , . t q C q q C n q q n q F f F W M f F M f F f    − + − +  = = = (v) By (4.3), (5.4) we have ( ) ( ) ( ) ( ) ( ) 1 1 2 , , 2 1 1 2 , 2 , # . t q q n q q n n q n q t t q q f g W M f M g W M f W M g f g        − − + + − − + +  =    =  =  (vi) By (3.6), (4.3) ,(5.4) we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , 2 1 , 2 2 , 1 2 , 2 , # . q q q n q q n n q n q n q t q q f g M M f M g M M f R g MR W M f g f g          − − + − + + − + +  =     =     =    =  This achieves the proof. 6. Generalized Wavelets Definition 6.1 A generalized q-Bessel wavelet is a function ,2, ,q n g L   satisfying the admissibility condition ( )( ) 2 , 0 0 . q g q d C F g       =   (6.1) Remark 6.1 By (3.6) and (6.1), ,2, ,q n g L   is a generalized q-Bessel wavelet if and only if, 1 M g − is a q-Bessel wavelet of order 2n + , and we have ( )( ) 1 2 1 2 , 2 0 . q n g q n M g d C F M g C      −  − + + = = (6.2) Note 6.1 For ,2, ,q n g L   where qa +  and  0qb +   we have ( ) ( )( )1/ 2, , , , ,a b n q b n ag x a T g x   = (6.3) where a g is given in (2.12) and ,q bT  are the generalized translation operators defined by (4.1). 14 Int. J. Anal. Appl. (2022), 20:33 Theorem 6.1 For all q a +  and  0qb +   we have ( ) ( ) ( ) ( ) 22 1 , , , , nn a b n a b g x bx M g x   + − = (6.4) Proof. Using (2.11), (4.1) and (6.3) we have ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1/ 2 , , , , , 2 1/ 2 2 1 , 2 1/ 2 2 1 , 22 1 , , a b n q b n a n n a b a n n q b a nn q b g x a T g x bx a M g x bx a M g x bx M g x        + − + − + − = = = = which ends the proof. Definition 6.2 Let  ( ),2, , 0q n qg L  +   be a generalized a q-Bessel wavelet. Then for a function  ( ),2, , 0q n qf L  +   , the continuous generalized a q-Bessel wavelet transform by ( )( ) ( ) ( )  2 1, , , 2 , , , 0 , , 0 , q g n q n a b n q q q f a b c f x g x x d x a b       + + + + =      (6.5) where ( ) ( )1/ 2, , , , ,a b n q b n ag x a T g   = and ( ) 2 2 1 / a g g x a a + = . It can also be written in the form ( )( ) ( )1/ 2, , , # ,q g n q af a b a f g b   = (6.6) where # q is the generalized convolution product given by (4.2). Theorem 6.2 We have ( )( ) ( ) ( )( )1 2 2 1 , , , , , . n n q g n q M g f a b b S M f a b    − + − = (6.7) Proof. From (2.10), (6.4) and (6.5) we deduce that ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )1 2 1 , , , 2 , , , 0 22 1 2 2 1 , 2 , 0 22 1 1 2 4 1 , 2 , 0 2 2 1 , , , , q g n q n a b n q nn n q n q a b nn n q n q a b n n q M g f a b c f x g x x d x c f x b M g x x x d x c b M f x M g x x d x b S M f a b             −  + +  + − + +  + − − + + + + − = = = =    15 Int. J. Anal. Appl. (2022), 20:33 which concludes the proof. Theorem 6.3 (Plancherel formula) Let  ( ),2, , 0q n qg L  +   be a generalized wavelet. For every  ( ),2, , 0q n qf L  +   we have the Plancherel formula ( ) ( )( ) 22 2 1 2 1 , , 2 0 0 0 1 , q q q g n q g d a f x x d x f a b b d b C a        + + =  . Proof. By (6.2) and Theorem 2.1(i) we have ( )( ) ( )( ) ( ) ( ) 1 1 22 2 1 2 1 2 4 1 , , 2 2, 0 0 0 0 2 2 1 2 4 1 0 2 2 1 0 , , . q qn n q g n q qq M g n n qM g g q d a d a f a b b d b S M f a b b d b a a C M f x x d x C f x x d x         − −     + + − + +  + − + +  + = = =     Theorem 6.4 (Calderon’s formula) Let ,2, ,q n g L   be a generalized wavelet, such that ( ), , q q F g    . Then for ,2, ,q n f L   and 0    , the function ( ) ( )( ) ( ), 2 1, , , , , 2 0 1 , q q g n a b n q g d a f x f a b g x b d b C a         +  =  belongs to ,2, ,q n L  . Proof. By (6.2), (6.4), (6.7) and theorem 2.1(ii) we have ( )( ) ( ) ( )( )( ) ( ) 1 1 2 1 1 2 4 12 2 1 2, , , , , , 2 2 0 0 , ,1 , . qn nn q qq M g q g n a b n q n g M g d a d a S M f a b M g b d bx f a b g x b d b a C a C f x           − −   + − − + + + +    = =   Theorem 6.5 (Inversion formula) Let ,2, ,q n g L   be a generalized wavelet. If ,1, ,q n f L   and ( ), ,1, 2q q nF f L  + then we have ( ) ( )( ) ( ) 2 1, , , , , 2 0 0 1 , q q g n a b n q g d a f x f a b g x b d b C a       +   =       for 0x  . Proof. By (6.2), (6.4), (6.7 we have 16 Int. J. Anal. Appl. (2022), 20:33 ( )( ) ( ) ( )( )( )1 1 2 2 1 2 1 1 2 4 1 , , , , , 2 2 2, 0 0 0 0 1 , , , n q qn n q g n a b n q qn q M g g M g d a d ax f a b g x b d b S M f a b M g b d b C a C a        − −     + + − − + + +     =            the result shows from theorem 2.1(iii). 7. Inversion of the Intertwining Operator t q  Through the Generalized Wavelet Transform To obtain inversion formulas or t q  involving generalized wavelets, we have to establish some preliminary lemmas. Lemma 7.1 Let ( ),1, , ,2, ,0 [0, [,q n q ng L L dx     such that ( ) ( ),1, , [0, [,c q nF g L dx  and satisfying 2n   + such that ( )( ) ( )ncF g O = (7.1) as 0 → . Then , ,2, ,q g q n L    and ( )( ) ( )( ) ( )( ) 22 4 1 , 2 4 1 2 2 1 . n c q g cn n F F g        + + + +  + + = (7.2) Proof. We have ( ) ( )( ) ( ) 0 2 cos . c g x F g x d     =  So by (5.3), ( ) ( ) ( ) ( )2 0 , q n g x h x d         + =  where ( ) ( )( ) ( )( ) 22 4 1 2 4 1 2 2 1 n cn n h F g       + + + +  + + = Clearly, h ( ),1, , [0, [,q nL dx  . So by (7.2) and Theorem 6.3 we have ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) 2 22 4 1 2 0 0 1 22 4 1 0 1 1 2 , , , , n n c n c h d m n F g d m n F g d m n I I                  − − − +  − − − =   = +    = +     17 Int. J. Anal. Appl. (2022), 20:33 where ( ) ( )( ) 24 1 2 , 4 4 1 n m n n     + + − =  + + . By (7.1) there is a positive constant k such that ( ) 2 2 4 1 1 0 . 2 2 n k I k d n        − − −  =   − −  From the Plancherel theorem for the cosine transform, it follows that ( )( ) ( )( ) ( ) 2 2 22 4 1 2 1 0 0 , 2 n c c I F g d F g d g x dx           − − − =  =     which achieves the proof. Lemma 7.2 Let ( ),1, , ,2, ,0 [0, [,q n q ng L L dx     such that ( ) ( ),1, , [0, [,c q nF g L dx  and satisfying 2 4 1n  + + such that ( )( ) ( )cF g O   = (7.3) as 0 → . Then ,2, ,q q n g L    is a generalized wavelet and ( ) ( ), , , , [0, [,c q g q nF L dx   . Proof. By (7.3) and Lemma 7.1 , ,2, ,q q n g L    , ( ),q gF  is bounded and ( )( ) ( )2 4 1, 0. n q g F O as       − − −  = → Hence q g satisfies the admissibility condition (6.1). The continuous wavelet transform on  )0, is defined by ( )( ) ( ) ( )( ), 0 1 , , q g b a W f a b f x g x dx a   =  (7.4) where 0, 0a b  and ( ),2, , [0, [,q ng L dx  is a classical wavelet on  )0, , i.e., satisfies the admissibility condition ( ) ( )( ) 2 0 0 . q c d C g F g      =   (7.5) Remark 7.1 (ii) By (5.5), (6.1) and (7.5), ( )g D is a generalized wavelet, if and only if , t q g  is a wavelet and ( ), t q g g C C = . Lemma 7.3 Let g be as in Lemma 7.2. Then , , ,q p n f L    , p=1 or 2, we have 18 Int. J. Anal. Appl. (2022), 20:33 ( )( ) ( )( ) ( ) , ,2 4 1 1 , , . q g t q q g qn f a b W f a b a      + +  =    Proof. By (6.6) we have ( )( ) ( ) ( ) , ,2 4 1 1 , . q g q q gn a f a b f b a     + + =  But ( ) ( ), 2 1 q g q ana g a  = by (2.12) and (5.1). So by (5.6) and (7.4) we get ( )( ) ( ) ( ) ( ) ( )( ) ( ) , 2 4 1 2 4 1 ,2 4 1 1 , 1 1 , , q g q q an t q q an t q q g qn f a b f g b a f g b a W a b a           + + + + + +  =     =     =    which completes the proof. Theorem 7.1 Let g be as in Lemma 7.2. Then we have the following inversion formulas for t q  : (i) If ,1, ,q n f L   and ( ), ,1, 2q q nF f L  + then for almost all 0x  we have ( ) ( )( ) ( ) ( ) ( ) , 2 1 , , 2 4 2, 0 0 1 , . q g t q q g q q g na b da f x W f a b x b db C a         + + +    =         (ii) For ,1, , ,2, ,q n q n f L L     and 0    , the function ( ) ( )( ) ( ) ( ) ( ) , , 2 1 , , 2 4 2, 0 1 , q g t q q g q q g na b da f x W f a b x b db C a           + + +   =     satisfies , ,2, ,0, lim 0. q n f f    → → − = Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. 19 Int. J. Anal. Appl. 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