International Journal of Analysis and Applications Volume 18, Number 4 (2020), 531-549 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-531 CONTINUOUS AND DISCRETE WAVELET TRANSFORMS ASSOCIATED WITH HERMITE TRANSFORM C. P. PANDEY∗ AND PRANAMI PHUKAN 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 accomplished the concept of continuous and discrete Hermite wavelet trans- forms. We also discussed some basic properties of Hermite wavelet transform. Inversion formula and Parsevals formula for continuous Hermite wavelet transform is established. Moreover the discrete version of wavelet transform is discussed. 1. Introduction Many authors have defined wavelet transforms associated with different integral transforms. In ( [6], [5]) Pathak and Dixit, Pathak and Pandey defined the wavelet transform which are associated with the Hankel and Laguree transform respectively. In [7] Upadhyay and Tripathi defined continuous wavelet transform corresponding to Watson transform. In 2017 Prasad and Mandal [4] studied the Kontorovich-Lebedev wavelet transform and derived many important properties related to the KL-wavelet transform. In [1] Pathak and Abhishek studied the continuous and discrete wavelet transform associated with index Whittaker transform. Hans-Jurgen Glaeske [3] defined the translation and convolution operator associated with Hermite transform and proved so many important results related to these operator. Now, however to best our knowledge wavelet Received March 24th, 2020; accepted April 22nd, 2020; published May 11th, 2020. 2010 Mathematics Subject Classification. 42C40, 65R10, 44A35. Key words and phrases. Hermite transforms; continuous Hermite wavelet transform; discrete Hermite wavelet transform; Hermite convolution. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 531 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-531 Int. J. Anal. Appl. 18 (4) (2020) 532 associated with the Hermite transform is not defined. So we are interested to define the wavelet associated to Hermite transform and study the continuous as well as discrete wavelet transforms associated with this. The wavelet transform [8] of the function f ∈ L2(R) with respect to the wavelet φ ∈ L2(R) is defined by (Wφf) (ρ,σ) = ∫ ∞ −∞ f(t)φρ,σ(t)dt,ρ ∈ R,σ > 0, (1.1) where φρ,σ(t) = σ −1 2 φ ( t−ρ σ ) . (1.2) In terms of translation τρ defined by τρφ(t) = φ(t−ρ),ρ ∈ R, and dilation Dσ is defined by Dσφ(t) = σ −1 2 φ ( t σ ) ,σ > 0, we can write φρ,σ(t) = τρDσφ(t). (1.3) From equation 1.1 and 1.3 it is clear that wavelet transform of the function f on R is an integral transform for which the kernel is the dilated translate of φ. We can also express equation 1.1 as the convolution (Wφf) (ρ,σ) = (f ∗g0,σ) (ρ), (1.4) where g(t) = φ(−t). Since associated with each integral transform there exists a special kind of convolution, one can construct wavelet transform corresponding to an integral transform using the associated convolution. We construct wavelet and wavelet transform on the interval (−∞,∞) by using the theory of Hermite transforms [2] and associated convolution involving the function H(µ)n (x) = exp ( −x2 2 ) H̃(µ)n (x),x ∈ R, where H̃ (µ) n (x) is the normalized Hermite polynomial, where α > −1, is given by H̃(µ)n (x) =   H (µ) 2k (x) H (µ) 2k (0) = R (µ−12 ) k (x 2),n = 2k H (µ) 2k+1 (x)( H (µ) 2k+1 (x) ) ′(0) = xR (µ+ 12 ) k (x 2),n = 2k + 1 , Int. J. Anal. Appl. 18 (4) (2020) 533 and H(µ)n (x) =   (−1)k22kk!L( µ−1 2 ) k (x 2),n = 2k (−1)k22k+1k!xL( µ+ 1 2 ) k (x 2),n = 2k + 1 . Set dµ(x) = e−x 2 |x|2µdx. (1.5) Let us consider the measurable function f(x) on the interval (−∞,∞). Then the Hermite transform is defined by H[f](n) = f̂(n) = ∫ ∞ −∞ f(x)H̃n(x)dµ(x),n ∈ N. (1.6) The inverse Hermite transform defined by f(x) = ∞∑ n=0 f̂(n)H̃(µ)n (x) [ h̃(µ)n ]−1 . (1.7) where h(µ)n = 2 2nΓ ([n 2 ] + 1 ) Γ ([ (n + 1) 2 ] + µ + 1 2 ) . Let the space of those real measurable functions f on (−∞,∞) be Lp,µ(−∞,∞), 1 ≤ p < ∞, for which ‖f‖p,µ = { ∫ ∞ −∞ | f(x) |p dµ(x)} 1 p ,p < ∞. (1.8) ‖f‖p,µ = esssupx∈R | f(x) |,p = ∞. (1.9) An inner product on L2,µ, is defined by 〈f,g〉 = ∫ ∞ −∞ f(x)g(x)dµ(x). (1.10) 2. Hermite translation and convolution In this section, Hermite translation and associated convolution will be discussed. To define the Hermite convolution ’*’ we have to introduce Hermite translation. For this purpose we need the basic function K (µ) H (x,y,z) ∼ ∞∑ n=0 [ h̃(µ)n ]−1 H̃(µ)n (x)H̃ (µ) n (y)H̃ (µ) n (z). (2.1) Hence by equation 1.6 and 1.7, we have∫ ∞ −∞ K (µ) H (x,y,z)H̃ (µ) n (z)dµ(z) = H̃ (µ) n (x)H̃ (µ) n (y). (2.2) Clearly K (µ) H (x,y,z) is symmetric in x,y and z. Setting n = 0 in equation 2.2, we have∫ ∞ −∞ K (µ) H (x,y,z)dµ(z) = 1. (2.3) Int. J. Anal. Appl. 18 (4) (2020) 534 The Hermite translation τy of f ∈ Lp,µ(−∞,∞), 1 ≤ p < ∞ is defined τyf(x) = f(x,y) = ∫ ∞ −∞ f(z)K (µ) H (x,y,z)dµ(z), 1 ≤ p < ∞. (2.4) Lemma 2.1. For f ∈ Lp,µ and 1 ≤ p < ∞, ‖τ(µ)y f‖p,µ ≤‖f‖p,µ, (2.5) and the map: f → τyf is continuous and linear in Lp,µ. Proof. Proof is referred from [3]. � Let p,q,r ∈ (−∞,∞) and 1 r = 1 p + 1 q − 1. Then the Hermite convolution [3] of f ∈ Lp,µ(−∞,∞) and g ∈ Lq,µ(−∞,∞) is defined by following equation (f ∗g) (y) = ∫ ∞ −∞ τ(µ)y (f; x) g(x)dµ(x). (2.6) By using the relation defined in equation 2.4, convolution (f ∗g) can be defined as (f ∗g) (x) = ∫ ∞ −∞ ∫ ∞ −∞ f(z)g(x)dm(µ)x,y(z) = ∫ ∞ −∞ ∫ ∞ −∞ f(z)g(x)K (µ) H (x,y,z)dµ(x)dµ(z). (2.7) Also recall the following Lemma from [3]. Lemma 2.2. Let p,q,r ∈ (−∞,∞) and 1 r = 1 p + 1 q − 1,f ∈ Lp,µ(−∞,∞) and g ∈ Lq,µ(−∞,∞). Then the convolution (f ∗g) defined by equation 2.7 satisfies the following norm inequality: (i)‖f ∗g‖r,µ ≤‖f‖p,µ‖g‖q,µ. (2.8) Moreover f,g ∈ L2,µ, we get (ii) (f ∗g)∧ (n) = f̂(n)ĝ(n). (2.9) Lemma 2.3. For any f ∈ L2,µ the following Parseval Identity holds for Hermite transform:∑ n [ h̃(µ)n ]−1 | f̂(n) |2= ‖f‖22,µ. (2.10) Proof. Proof is referred from theorem 1 in ref. [2]. � For any f1,f2 ∈ L2,µ(−∞,∞) the below Parseval Identity holds for Hermite transform. See ref. [2].∑ n [ h̃(µ)n ]−1 f1(n)f2(n) = ∫ ∞ −∞ f1(x)f2(x)dµ(x) and ∑ n [ h̃(µ)n ]−1 f1(n)f2(n) = ∫ ∞ −∞ H−1[f1(n)][f2(n)]dµ(x). Int. J. Anal. Appl. 18 (4) (2020) 535 In this paper, following the technique of Pathak and Dixit [6] and Trimeche [9], Hermite wavelet transform is defined. The continuity and boundedness properties of Hermite wavelet transform is derived. A semi dis- crete Hermite wavelet transform is defined. Furthermore discrete Hermite wavelet transform is investigated. Using discrete Hermite wavelet, frame and Riesz basis [8] are also studied. 3. Continuous Hermite Wavelet Transform For a function φ ∈ Lp,µ(−∞,∞), defined the dilation Dσ by Dσφ(t) = φ(σt),σ > 0. (3.1) Using the Hermite translation 2.4 and above dilation, the Hermite wavelet φρ,σ(t) is defined as follows: φρ,σ(t) = τρDσφ(t) = τρφ(σt) (3.2) = ∫ ∞ 0 φ(σz)K (µ) H (ρ,t,z)dµ(z). (3.3) where ρ ≥ 0 and σ > 0. The integral is convergent by virtue of inequality 2.5. Definition 3.1. Admissible Hermite wavelet The function φ(·) ∈ Lp,µ(−∞,∞) is said to be admissible Hermite wavelet if φ(·) satisfies the following admissibility condition Cφ = ∞∑ n=0 | φ̂(n) |2 | n | < ∞, where φ̂(n) is the Hermite transform of φ. Continuous Hermite wavelet transform Using the wavelet φρ,σ we now define the continuous Hermite wavelet transform. ( H̃ (µ) φ f ) (ρ,σ) = 〈f(t),φρ,σ(t)〉 = ∫ ∞ −∞ f(t)φρ,σ(t)dµ(t) (3.4) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)K (µ) H (ρ,t,z)dµ(z)dµ(t) (3.5) provided the integral is convergent. Since by inequality 2.5 and definition φρ,σ ∈ Lp,µ whenever φ ∈ Lp,µ. By virtue of Lemma 2.2, the integral 3.5 is convergent for f ∈ Lq,µ, 1p + 1 q = 1. The Hermite wavelet transform can be expressed in the form of Hermite transform as follows. H [( H̃ (µ) φ f ) (ρ,σ) ] = f̂(n)φ̂(σ,n). Int. J. Anal. Appl. 18 (4) (2020) 536 Also the Hermite wavelet transform can be written as ( H̃ (µ) φ f ) (ρ,σ) = (f ∗φ(σ, ·)) (ρ). The continuity and boundedness results follow from the following theorem. Theorem 3.1. Let f(·) ∈ Lp,µ and φ(·) ∈ Lq,µ,σ > 0 with 1 ≤ p,q < ∞ and 1p + 1 q = 1. and ( H̃ (µ) φ f ) (ρ,σ) be continuous Hermite wavelet transform 3.5. Then (i) ‖ ( H̃ (µ) φ f ) (ρ,σ)‖r,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ, 1r = 1 p + 1 q − 1, 1 ≤ p,q,r < ∞. (ii) ‖ ( H̃ (µ) φ f ) (ρ,σ)‖∞,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ, 1p + 1 q = 1. Proof. (ii) Using representation 3.5, we have ( H̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)K (µ) H (ρ,t,z)dµ(z)dµ(t) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)K (µ) 1 p H (ρ,t,z)K (µ) 1 q H (ρ,t,z)dµ(z)dµ(t) using Holder’s inequality, we get | ( H̃ (µ) φ f ) (ρ,σ)| ≤ (∫ ∞ −∞ ∫ ∞ −∞ | f(t) |p K(µ)H (ρ,t,z)dµ(z)dµ(t) )1 p × (∫ ∞ −∞ ∫ ∞ −∞ | φ(σz) |q K(µ)H (ρ,t,z)dµ(z)dµ(t) )1 q = (∫ ∞ −∞ | f(t) |p dµ(t) ∫ ∞ −∞ K (µ) H (ρ,t,z)dµ(z) )1 p × (∫ ∞ −∞ | φ(σz) |q dµ(z) ∫ ∞ −∞ K (µ) H (ρ,t,z)dµ(t) )1 q by using equation 2.3, it follows that | ( H̃ (µ) φ f ) (ρ,σ)| ≤ ‖f‖p,µ‖φ(σ, ·)‖q,µ; so that ‖ ( H̃ (µ) φ f ) (ρ,σ)‖∞,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ. the inequality 1.1 follows from inequality 2.8. � Theorem 3.2. If φ is a basic Hermite wavelet and Ψ is any bounded function, then (φ∗Ψ) is also a hermite wavelet. Int. J. Anal. Appl. 18 (4) (2020) 537 Proof. C(φ∗Ψ) = ∞∑ n=0 |(φ∗ Ψ)∧(n)|2 n = ∞∑ n=0 |(φ)∧(n)(Ψ)∧(n)|2 n ≤ |(Ψ)∧(n)| ∞∑ n=0 |(φ)∧(n)|2 n < ∞ Hence (φ∗ Ψ) is a Hermite wavelet. � Basic Properties of Continuous Hermite Wavelet Transform Theorem 3.3. Let φ and Ψ be two wavelets and f,g be two functions belong to Lp,µ(−∞,∞), then (i) Linearity property: H (µ) φ (ηf + ζg)(σ,ρ) = ηH (µ) φ (f)(σ,ρ) + ζH (µ) φ (g)(σ,ρ) where η and ζ are any two scalars. (ii) Shift property ( H (µ) φ f ) (x− τ)(σ,ρ) = ( H (µ) φ f ) (σ,ρ− τ) where τ is any scalar. (iii) Scaling property If c 6= 0 is any scalar, then the Hermite wavelet transform of the scaled function fc(x) = 1 c f ( 1 2 ) is ( H (µ) φ fc ) (ρ,σ) = H (µ) φ f (σ c , ρ c ) (iv) Symmetry property: ( H (µ) φ f ) (σ,ρ) = ( H (µ) φ f ) (φ) ( 1 σ , −1 ρ ) (v) Parity property ( H (µ) pφ pf ) (σ,ρ) = ( H (µ) φ f ) (σ,−ρ) where p is the parity operator defined by pf(x) = f(−x). Proof. The proof is the straight forward application of Hermite transform. � Plancharel and Persevals relation for Continuous Hermite wavelet Transform Let f,g ∈ L2,µ(−∞,∞) and φ1,φ2 ∈ L2,µ(−∞,∞) are two Hermite wavelets. Then we have 〈 ( H (µ) φ1 f ) (σ,ρ), ( H (µ) φ2 g ) (σ,ρ)〉L2,µ((−∞,∞)×(−∞,∞)) = cφ1,φ2〈f,g〉L2,µ(−∞,∞), (3.6) Int. J. Anal. Appl. 18 (4) (2020) 538 where cφ1,φ2 = ∫ ∞ 0 φ1(σ,n)φ2(ρ,n)dµ(σ). Proof. Let f,g ∈ L2,µ(−∞,∞) then from 3.4, we have∫ ∞ 0 ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ) ( H (µ) φ g ) (σ,ρ)dµ(σ)dµ(ρ) = ∫ ∞ 0 ∫ ∞ −∞ H (µ)−1 φ [ f̂(n)φ1(σ,n) ] (ρ) H (µ)−1 φ [ĝ(n)φ2(σ,n)] (ρ)dµ(σ)dµ(ρ) now by using 2.10 we get ∫ ∞ 0 ∫ ∞ −∞ f(x)φ1(σ,x)(ρ)g(x)φ2(σ,x)(ρ)dµ(σ)dµ(ρ) = ∑ n [ h̃(µ)n ]−1 f̂(n)ĝ(n) ∫ ∞ −∞ φ1(σ,n)φ2(σ,n)dµ(σ) (3.7) = cφ1,φ2 ∑ n [ h̃(µ)n ]−1 f̂(n)ĝ(n). Hence by using the Parseval formula for Hermite transform, we get∫ ∞ 0 ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ) ( H (µ) φ g ) (σ,ρ)dµ(σ)dµ(ρ) = cφ1,φ2f̂(n)ĝ(n) = cφ1,φ2〈f,g〉L2,µ(−∞,∞). (3.8) � Theorem 3.4. (Inversion formula) Let f ∈ L2,µ(−∞,∞) and φ is Hermite wavelet defines continuous Hermite wavelet transform. Then, f(x) = 1 cφ ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ), where cφ is the Admissible Hermite wavelet. Proof. Let h(x) ∈ L2,µ(−∞,∞) be any function, then by applying previous theorem, we have cφ〈f,h〉L2,µ(−∞,∞) = ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ) ( H (µ) φ h ) (σ,ρ)dµ(σ)dµ(ρ) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ) ∫ ∞ −∞ h(t)φρ,σ(t)dtdµ(σ)dµ(ρ) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ)φρ,σ(t)h(t)dtdµ(σ)dµ(ρ) = ∫ ∞ −∞ g(t)h(t)dt = 〈g,h〉, (3.9) Int. J. Anal. Appl. 18 (4) (2020) 539 where, g = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ). Then, cφ〈f,h〉 = 〈g,h〉 f = 1 cφ g = 1 2cφ ∫ ∞ −∞ ∫ ∞ −∞ ( H (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ). If f = h, ‖f‖2L2,µ(−∞,∞) = ∫ ∞ −∞ ∫ ∞ −∞ | ( H (µ) φ f ) (σ,ρ) |2 dµ(σ)dµ(ρ). Moreover the Hermite wavelet transform is isometry from L2,µ(−∞,∞) to L2,µ(−∞,∞)×L2,µ(−∞,∞). � A General Reconstruction Formula In this section, we show that the function f can be recovered from its Hermite wavelet transform. In derived the reconstruction formula, we need the following lemma. Lemma 3.1. Let f ∈ L2,µ and φ ∈ L2,µ be a basic wavelet, which defines Hermite wavelet transform 3.5. Then ( H̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n), (3.10) where φ̂(σ,n) = ∫ ∞ −∞ φ(σz)H̃(µ)n (z)dµ(z). (3.11) Proof. Using representation 3.5, we have( H̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)K (µ) H (ρ,t,z)dµ(z)dµ(t) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)dµ(z)dµ(t) ( ∞∑ n=0 [ h̃(µ)n ]−1 H̃(µ)n (ρ)H̃ (µ) n (t)H̃ (µ) n (z) ) = ∞∑ n=0 [ h̃(µ)n ]−1 H̃(µ)n (ρ) (∫ ∞ −∞ f(t)H̃(µ)n (t)dµ(t) ∫ ∞ −∞ φ(σz)H̃(µ)n (z)dµ(z) ) = ∞∑ n=0 [ h̃(µ)n ]−1 H̃(µ)n (ρ)f̂(n)φ̂(σ,n) = ( f̂(n)φ̂(σ,n) )∨ (ρ). ∴ ( H̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n). This completes the proof. � Int. J. Anal. Appl. 18 (4) (2020) 540 Theorem 3.5. Let f ∈ L2,µ and φ be a basic wavelet which defines Hermite wavelet transform by equation 3.5. Let q(σ) > 0 be a weight function such that Q(n) = ∫ ∞ 0 q(σ) | φ̂(σ,n) |2 dµ(σ) > 0. (3.12) Set φ̂ρ,σ(n) = φ̂ρ,σ(n) Q(n) . (3.13) Then f(t) = ∫ ∞ 0 ∫ ∞ −∞ q(σ) ( H̃ (µ) φ f ) (ρ,σ)φb,a(t)dµ(σ)dµ(ρ). (3.14) Proof. From equation 3.10, we have ( H̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n) ⇒ ∫ ∞ −∞ ( H̃ (µ) φ f ) (ρ,σ)H̃ (µ) φ (b)dµ(b) = f̂(n)φ̂(σ,n). multiplying both sides by φ̂(σ,n) and weight function q(σ) and integrating with respect to dµ(σ), we have∫ ∞ 0 q(σ)φ̂(σ,n) (∫ ∞ −∞ ( H̃ (µ) φ f ) (ρ,σ)H̃(µ)n (ρ)dµ(ρ) ) dµ(σ) = ∫ ∞ 0 q(σ)f̂(n)φ̂(σ,n)φ̂(σ,n)dµ(σ) ⇒ ∫ ∞ 0 q(σ)φ̂(σ,n) (∫ ∞ −∞ ( H̃ (µ) φ f ) (ρ,σ)H̃(µ)n (ρ)dµ(ρ) ) dµ(σ) = ∫ ∞ 0 q(σ)f̂(n) | φ(σ,n) |2 dµ(σ). (3.15) Equation 3.11 and 3.15 gives f̂(n)Q(n) = ∫ ∞ 0 q(σ)φ̂(σ,n)dµ(σ) ∫ ∞ −∞ ( H̃ (µ) φ f ) (ρ,σ)H̃(µ)n (ρ)dµ(ρ) f̂(n) = 1 Q(n) ∫ ∞ 0 q(σ) ∫ ∞ −∞ ( H̃ (µ) φ f ) (ρ,σ)φ̂(σ,n)H̃(µ)n (ρ)dµ(σ)dµ(ρ) (3.16) We also have from equation 3.3, φρ,σ(t) = ∫ ∞ −∞ φ(σz) ∞∑ n=0 [ h(µ)n ]−1 H̃(µ)n (ρ)H̃ (µ) n (t)H̃ (µ) n (z)dµ(z) = ∞∑ n=0 [ h(µ)n ]−1 H̃(µ)n (ρ)H̃ (µ) n (t) ∫ ∞ −∞ φ(σz)H̃(µ)n (z)dµ(z) = ∞∑ n=0 [ h(µ)n ]−1 H̃(µ)n (ρ)H̃ (µ) n (t)φ̂(σ,n) = ( φ̂(σ,n)H̃(µ)n (ρ) )∨ (t). Int. J. Anal. Appl. 18 (4) (2020) 541 Therefore φρ,σ(t) = ( φ̂(σ,n)H̃(µ)n (ρ) )∨ (t). φ̂ρ,σ(t) = φ̂(σ,n)H̃ (µ) n (ρ). (3.17) Using equation 3.17 in 3.16, we have f̂(n) = 1 Q(n) ∫ ∞ 0 q(σ) ∫ ∞ −∞ φ̂ρ,σ(n) ( H̃ (µ) φ f ) (ρ,σ)dµ(σ)dµ(ρ). (3.18) From equation 3.13 it follows that f̂(n) = ∫ ∞ 0 q(σ) ∫ ∞ −∞ φ̂ρ,σ(n) ( H̃ (µ) φ f ) (ρ,σ)dµ(σ)dµ(ρ). (3.19) From equation 1.7 and 3.19, we have f(t) = ∞∑ n=0 [ h(µ)n ]−1 H̃(µ)n (t) ∫ ∞ −∞ ∫ ∞ −∞ q(σ) ( H̃ (µ) φ f ) (ρ,σ)φ̂ρ,σ(n)dµ(σ)dµ(ρ) = ∞∑ n=0 q(σ) ( H̃ (µ) φ f ) (ρ,σ) ∞∑ n=0 [ h(µ)n ]−1 φ̂ρ,σ(n)H̃(µ)n (t)dµ(σ)dµ(ρ) = ∫ ∞ 0 ∫ ∞ −∞ q(σ) ( H̃ (µ) φ f ) (ρ,σ)φρ,σ(n)dµ(σ)dµ(ρ). This completes the proof of theorem 3.5. A characterization of φρ,σ is given below. � Theorem 3.6. Assume that there exist positive constant A and B such that, 0 < A ≤ Q(n) ≤ B < ∞ (3.20) Let φσ(t) = ∞∑ n=0 [ h (µ) n ]−1 Q(n) φ̂(σ,n)H̃(µ)n (t). (3.21) Then (i)φρ,σ(t) = τρφ σ(t); (3.22) (ii)‖φρ,σ‖2,µ ≤ A−1‖φρ,σ‖2,µ. (3.23) Int. J. Anal. Appl. 18 (4) (2020) 542 Proof. (i) Using equations 1.7, 2.2, 3.13 and 3.16, we have φρ,σ(t) = φρ,σ(n) Q(n) = φ(σ,n)H̃ (µ) n (ρ) Q(n) = ∑∞ n=0 φ̂(σ,n)H̃ (µ) n (t) [ h̃ (µ) n ]−1 H̃ (µ) n (ρ) Q(n) = ∑∞ n=0 φ̂ρ,σ(n) H̃ (µ) n (ρ) H̃ (µ) n (t) [ h̃ (µ) n ]−1 H̃ (µ) n (ρ) Q(n) = ∑∞ n=0 φ̂ ρ,σ(n)Q(n)H̃ (µ) n (t) [ h̃ (µ) n ]−1 Q(n) = ∞∑ n=0 φ̂ρ,σ(n)H̃ (µ) n (t) [ h̃(µ)n ]−1 = ∞∑ n=0 [ h̃(µ)n ]−1 φ̂ρ,σ(n) Q(n) H̃(µ)n (t) = ∞∑ n=0 [ h̃ (µ) n ]−1 Q(n) φ̂(σ,n)H̃(µ)n (ρ)H̃ (µ) n (t) = ∞∑ n=0 [ h̃ (µ) n ]−1 Q(n) φ̂(σ,n) (∫ ∞ −∞ K (µ) H (x,y,z)H̃ (µ) n (z)dµ(z) ) = ∫ ∞ −∞ K (µ) H (x,y,z)   ∞∑ n=0 [ h̃ (µ) n ]−1 Q(n) φ̂(σ,n)H̃(µ)n (z)  dµ(z) = ∫ ∞ −∞ φσK (µ) H (x,y,z)dµ(z) = τρφ σ(z), where φσ(t) is given in equation 3.21. (ii) From equation 3.13, we have | φ̂ρ,σ |≤ A−1 | φρ,σ(n) |; (3.24) so that ∞∑ n=0 [ h(µ)n ]−1 | φ̂ρ,σ(n) |2≤ A−2 ∞∑ n=0 [ h(µ)n ]−1 | φρ,σ(n) |2 . Using equation 2.10, we get ‖φρ,σ‖2,µ ≤‖φρ,σ‖2,µ. � Int. J. Anal. Appl. 18 (4) (2020) 543 4. The Discrete Hermite Wavelet Transform The continuous Hermite wavelet transform of the function f in terms of two continuous parameters σ and ρ can be converted into a semi-discrete Hermite wavelet transform by assuming that σ = 2−j,j ∈ Z and ρ ∈ R+. In what follows we assume that φ ∈ L1,µ ∩L2,µ satisfies the so called ’stability condition’. A ≤ ∞∑ j=−∞ | φ̂(2−jn) |2≤ B a.e. (4.1) for certain positive constants A and B, −∞ < A ≤ B < ∞. The function φ ∈ L1,µ ∩ L2,µ satisfying condition 4.1 is called dyadic wavelet. Using definition 3.4, we define the semi discrete Hermite wavelet transform of any f ∈ L1,µ ∩L2,µ by ( H φ j f ) (ρ) = ( H φ j f ) (ρ, 2−j) (4.2) = 〈f(t),φρ,2−j (t)〉 = ∫ ∞ −∞ f(t)φρ,2−j (t)dµ(t) = ∫ ∞ −∞ f(t)τρφ(2−jt)dµ(t) = ( f ∗φj ) (ρ), where φj(z) = φ(2 −jz),j ∈ Z. Theorem 4.1. Assume that the semi discrete Hermite wavelet transform of any f ∈ L1,µ ∩L2,µ is defined by the equation 4.2. Let us consider another wavelet φ∗ defined by means of its Hermite transform φ̂∗(n) = φ̂(n)∑∞ l→−∞ | φ̂(2−ln) |2 . (4.3) Then f(t) = ∞∑ j=−∞ ∫ ∞ −∞ ( H φ j f ) (ρ) ( φ̂∗(2−jn)H̃(µ)n (t) )∨ (ρ)dµ(ρ). (4.4) Proof. In view of relations 1.7, 3.23 and 2.9, Int. J. Anal. Appl. 18 (4) (2020) 544 ∞∑ j=−∞ ∫ ∞ −∞ ( H φ j f ) (ρ) ( φ̂∗(2−jn)H̃(µ)n (t) )∨ (ρ)dµ(ρ) = ∞∑ j=−∞ ∫ ∞ −∞ ( H φ j f ) (ρ) [ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)H̃(µ)n (t)H̃ (µ) n (ρ) ] dµ(ρ) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)H̃(µ)n (t) ∫ ∞ −∞ ( H φ j f ) (ρ)H̃(µ)n (ρ)dµ(ρ) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)H̃(µ)n (t) ( H φ j f )∧ (n) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)H̃(µ)n (t) ( f ∗φj )∧ (n) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)H̃(µ)n (t)f̂(n)φ̂(2 −jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)H̃(µ)n (t)φ̂ ∗(2−jn)φ̂(2−jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)H̃(µ)n (t) φ̂(2−jn)∑ l | φ̂(2−j2−ln) |2 φ̂(2−jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)H̃(µ)n (t) | φ̂(2−jn) |2∑ l | φ̂(2−j2−ln) |2 = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)H̃(µ)n (t) = f(t). The above theorem leads to the following definition of dyadic dual. � Definition 4.1. A function φ̃ ∈ L1,µ ∩ L2,µ, is called a dyadic dual of a dyadic wavelet φ, if every f ∈ L1,µ ∩L2,µ can be expressed as f(t) = ∑ j ∫ ∞ −∞ ( H φ j f ) (ρ) ( φ̃(2−jn)H̃(µ)n (t) )∨ (ρ)dµ(ρ). (4.5) So far we have considered semi-discrete Hermite wavelet transform of any f ∈ L1,µ ∩ L2,µ discretizing only variable a. Now, we discretize the translation parameter b also by restricting it to the discrete set of points: ρj,k = k 2j ρ0; j ∈ Z,k ∈ N, (4.6) where ρ0 > 0 is a fixed constant. We write, φρ0;j,k(t) = φρ,j,k;σj (t) = φ(2 −jt, 2−jkρ0). (4.7) Int. J. Anal. Appl. 18 (4) (2020) 545 Then the discrete Hermite wavelet transform of any f ∈ L2,µ can be expressed as( H (µ) φ f ) (ρj,k,σj) = 〈f,φρ0;j,k〉,j ∈ Z,k,n ∈ N. (4.8) The ’stability condition’ for this reconstruction takes the form A‖f‖22,µ ≤ ∑ j∈Z |〈f,φρ0;j,k〉| 2 ≤ B‖f‖22,µ,k ∈ N, (4.9) where A and B are positive constant such that 0 ≤ A ≤ B < ∞. Theorem 4.2. Assume that the discrete Hermite wavelet transform of any f ∈ L2,µ is defined by 4.8 and stability condition 4.9 holds. Let T be a linear operator on L2,µ defined by Tf = ∑ j∈Z,k∈N0 〈f,φρ0;j,k〉µφρ0;j,k. (4.10) Then f = ∑ 〈f,φρ0;j,k〉µφ j,k ρ0 , where, φj,kρ0 = T −1φρ0;j,k; j ∈ Z,k ∈ N0. Proof. From the stability condition 4.9, it follows that the operator defined by equation 4.10 is a one-one bounded linear operator. Set g = Tf,f ∈ L2,µ. Then from equation 4.10, we have 〈Tf,f〉 = ∑ j∈Z,k∈N0 | f,φρ0;j,k | 2 . Therefore, from condition 4.9, A‖T−1g‖22,µ = A‖f‖t 2 2 ≤ ∑ | 〈f,ψρ0;j,k〉 | 2= 〈Tf,f〉µ = 〈gT−1g〉µ ≤‖g‖2,µ‖T−1g‖2,µ by Schwartz equality. Therefore, ‖T−1g‖2,µ ≤ 1 A ‖g‖2,µ. Hence, every f ∈ L2,µ can be constructed from its discrete Hermite wavelet transform given by 4.8. Thus f = T−1Tf = ∑ j∈Z,k∈N0 〈f,φρ0;j,k〉T −1φρ0;j,k. (4.11) Int. J. Anal. Appl. 18 (4) (2020) 546 Finally, set φj,kρ0 = T −1φρ0;j,k; j ∈ Z,n ∈ N0. Then the construction 4.11 can be expressed as f = ∑ j∈Z,k∈N0 〈f,φρ0;j,k〉φ j,k ρ0 , which completes the proof of theorem 4.2. � Frames and Riesz basis in L2,µ In this section, using φρ0;j,k a frame is defined and Riesz basis of L2,µ is studied. Definition 4.2. A function f ∈ L2,µ is said to generate a frame {φρ0;j,k} of L2,µ with sampling rate ρ0 if condition 4.8 holds for some positive constant A and B. If A = B, then the frame is called a tight frame. Definition 4.3. A function f ∈ L2,µ is said to generate a Riesz basis of {φρ0;j,k} with sampling rate ρ0 if the following two properties are satisfied. (i) The linear span 〈φρ0;j,k; j ∈ Z〉 is dense in L2,µ. (ii) There exist positive constants A and B with 0 < A ≤ B < ∞ such that A‖cj,k‖2l2 ≤‖ ∑ j,k∈Z cj,kφρ0;j,k‖ 2 2,µ ≤ B‖cj,k‖ 2 l2 (4.12) for all {cj,k}∈ l2(N2). Here A and B are called the Riesz bounds of {φρ0;j,k}. Theorem 4.3. Let φ ∈ L2,µ, then the following statements are equivalent. (i) {φρ0;j,k} is a Riesz basis of L2,µ. (ii) {φρ0;j,k} is a frame of L2,µ and is also an l2 linearly independent family in the sense that if∑ j,k cj,kφρ0;j,k = 0 and {cj,k} ∈ l 2, then cj,k = 0. Furthermore, the Riesz bounds and frame bounds agree. Proof. It follows from property 4.12 that any Riesz basis is l2-linearly independent. Let {φρ0;j,k} be a Riesz basis with Riesz bounds A and B, and consider the matrix operator: M = [γl,m;j,k](l,m)(j,k)∈N×N , where the entries are defined by γl,m;j,k = 〈φρ0;l,m,φρ0;j,k〉µ. (4.13) Then from property 4.12, we have A‖{cj,k}‖2l2 ≤ ∑ l,m,j,k cl,mγl,m;j,kcj,k ≤ B‖{cj,k}‖2l2 ; Int. J. Anal. Appl. 18 (4) (2020) 547 so that M is positive definite. We denote the inverse of M by M−1 = [µl,m;j,k](l,m)(j,k)∈N×N , (4.14) which means that both ∑ r,s µl,m;r,sγr,s;j,k = δl,jδm,k; l,m,j,k ∈ N, (4.15) and B−1‖{cj,k}‖2l2 ≤ ∑ l,m,j,k cl,mµl,m;j,kcj,k ≤ A−1‖{cj,k}‖2l2, (4.16) are satisfied. This allows us to introduce φl,m(x) = ∑ j,k µl,m;j,kφρ0;j,k(x). (4.17) Clearly, φl,m ∈ L2,µ and it follows from equation 4.15 and 4.13 that 〈φl,m,φρ0;j,k〉µ = δl,jδm,k;l,m,j,k ∈ N, which means that {φl,m} is the basis of L2,µ, which is dual to {φρ0;j,k}. Furthermore from equation 4.15 and 4.17, we conclude that 〈φl,m,φj,k〉µ = µl,m;j,k and the Reisz bounds of {φl,m} are B−1 and A−1. In particular, for any f ∈ L2,µ we may write f(x) = ∑ j,k 〈f,φρ0;j,k〉µφ j,k(x) and B−1 ∑ j,k | 〈f,φρ0;j,k〉µ | 2≤‖f‖22,µ ≤ A −1 ∑ j,k | 〈f,φρ0;j,k〉µ | 2 . (4.18) Since condition 4.18 is equivalent to condition 4.8, therefore statement (i) implies statement (ii). To prove the converse part, we recall Theorem 3.5 and we have any g ∈ L2,µ and f = T−1g, g(x) = ∑ m∈Z,n∈N 〈f,φρ0;j,k〉µφρ0;j,k. Also by the l2-linear independence of {φρ0;j,k}, this representation is unique. From the Banach-Steinhaus and open mapping theorem it follows that {φρ0;j,k}, is a Riesz basis of L2,µ. � Int. J. Anal. Appl. 18 (4) (2020) 548 Example 4.1. Let the mother wavelet be φ(t) =   1, 0 ≤ t ≤ 1 2 −1, −1 2 ≤ t < 1 0, otherwise . (4.19) This mother wavelet is called Haar wavelet. This is piecewise continuous. Using this wavelet we have following expression for φ(σt). φ(σt) =   1, 0 ≤ t ≤ 1 2σ −1, 1 2σ ≤ t < 1 σ 0, otherwise (4.20) Let f(t) = t−2µe−2t. Then Hermite transform of f(t) is given by H [ t−2µe−2t ] = ∫ ∞ −∞ f(t)H̃(µ)n (t)dµ(t) = ∫ ∞ −∞ f(t)e−t 2 | t |2µ H̃(µ)n (t)dt = ∫ ∞ −∞ t−2µe−2te t2 2 H(µ)n (t) | t | 2µ e−t 2 dt = e − ( t2+2t−t 2 2 ) H(µ)n (t)dt = √ π(2σ)neα 2 (4.21) Now, ∫ ∞ −∞ φ(σz)H̃(µ)n (z)dµ(z) = ∫ 1 2σ 0 H̃(µ)n (z)dµ(z) − ∫ 1 σ 1 2σ H̃(µ)n (z)dµ(z) = 2 ∫ 1 σ 0 H̃(µ)n (z)dµ(z) − ∫ 1 2σ 0 H̃(µ)n (z)dµ(z) = 2φ1(n,µ) −φ2(n,µ), (4.22) where φ1(n,µ) = ∫ 1 σ 0 H̃ (µ) n (z)dµ(z) and φ2(n,µ) = ∫ 1 2σ 0 H̃ (µ) n (z)dµ(z). Using representation 3.5 and 2.1, we have( H̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)K (µ) H (ρ,t,z)dµ(z)dµ(t) = ∞∑ n=0 [ h̃(µ)n ]−1 H̃(µ)n (ρ) (∫ ∞ −∞ f(t)H̃(µ)n (z)dµ(z) )(∫ ∞ −∞ φ(σz)H̃(µ)n (z)dµ(z) ) . From equations 4.21 and 4.22, it follows that( H̃ (µ) φ f ) (ρ,σ) = ∞∑ n=0 22nΓ ([n 2 ] + 1 ) Γ ([ n + 1 2 ] + µ + 1 ) H̃(µ)n (ρ) √ π(2σ)neα 2 (2φ1(n,µ) −φ2(n,µ)). Int. J. Anal. Appl. 18 (4) (2020) 549 Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] A. Pathak, Abhishek, Wavelet transforms associated with the Index Whittaker transform, ArXiv:1908.03766 [Math]. 2019. [2] C. Markett, The Product Formula and Convolution Structure Associated with the Generalized Hermite Polynomials, J. Approx. Theory, 73 (1993) 199–217. [3] H. Glaeske, Convolution structure of (generalized) Hermite transforms, Algebra Analysis and related topics, Banach Center Publications, Vol 53 (1), 113-120 (2000). [4] A. 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