The fractional Hankel-type integral wavelet packet transformation

Abstract

The wavelet packet transformation involving the fractional powers of Hankel-type integral transformation is defined and discussed on its some basic properties. An inversion formula of this transformation is also obtained. Some examples are given.

1 Introduction

Let $L^p_{\mu ,\alpha , \nu }(I)$ be the space of all real valued measurable functions $\psi$ defined on $I=(0,\infty )$ for which $\int _0^\infty |\psi (t)|t^{\nu \mu -\alpha +2\nu -1} dt$ exist. Also the space $L^\infty (I)$ be the collection of almost everywhere bounded integrable functions. Hence the norm is defined as

$$\begin{aligned} \Vert \psi \Vert _{L^p_{\mu , \alpha , \nu }(I)} = {\left\{ \begin{array}{ll} \Big (\int ^{\infty }_0 \mid \psi (t)\mid ^{p} t^{\nu \mu -\alpha +2\nu -1}dt\Big )^\frac{1}{p}, & 1< p < \infty , \quad \mu ,\alpha ,\nu \in \mathbb {R}\\ ess \sup _{t\in I}\mid \psi (t)\mid , & p = \infty . \end{array}\right. } \end{aligned}$$

(1)

The fractional powers of $\theta ~~(0<\theta <\pi )$ of Hankel-type integral transformation $H_{\mu ,\alpha ,\beta ,\nu }^{\theta }$ of functions $f \in L^p_{\mu ,\alpha , \nu }(I)$ depending upon three real parameters $(\alpha ,\beta ,\nu )$ is defined as [6, 7]

$$\begin{aligned} (H_{\mu ,\alpha ,\beta ,\nu }^{\theta }f)(w)=\hat{f}_{\mu ,\alpha ,\beta ,\nu }^{\theta }(w)= \int _{0}^{\infty } K^{\theta }(t,w)f(t) dt \end{aligned}$$

(2)

where

$$\begin{aligned} K^{\theta }(t,w)= {\left\{ \begin{array}{lll} \nu \beta C^\theta _{\mu ,\alpha }(tw)^{\alpha } J_\mu \big (\beta (tw)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+w^{2\nu }) \cot \theta }t^{-1-2\alpha +2\nu } ,&\theta \ne n\pi \\ \nu \beta (tw)^{\alpha } J_\mu \big (\beta (tw)^{\nu }\big )t^{-1-2\alpha +2\nu } ,& \theta = \frac{\pi }{2}\\ \delta (t-w),&\theta = n \pi \end{array}\right. } \end{aligned}$$

$\forall ~~~~ n\in \mathbb {Z}$, $\nu \mu +2\nu -\alpha \ge 1$, $C^\theta _{\mu ,\alpha }= \frac{e^{i(1+\mu )(\theta -\frac{\pi }{2})}}{\sin \theta }$ and $J_\mu$ is the Bessel function of first kind of order $\mu$.

The inverse of (1) is defined as follows:

$$\begin{aligned} f(t)=\Big (H^{-\theta }_{\mu ,\alpha ,\beta ,\nu }\hat{f}_{\mu ,\alpha ,\beta ,\nu }^{\theta }\Big )(t)=\int _{0}^{\infty }K^{-\theta }(w,t)\hat{f}_{\mu ,\alpha ,\beta ,\nu }^{\theta }(w) dw, \end{aligned}$$

(3)

where $\delta$ is Dirac-delta function and it is given by

$$\begin{aligned} \delta (t - w) = \nu ^2 \beta ^2 t^\alpha w^{-1-\alpha +2\nu } \int _o^\infty y^{-1+2\nu } J_{\mu }\left( \beta (ty)^\nu \right) J_{\mu }\left( \beta (wy)^\nu \right) dy. \end{aligned}$$

(4)

The Parseval’s identity becomes

$$\begin{aligned} \int _{0}^{\infty }t^{-1-2\alpha +2\nu } f(t)\overline{g(t)}dt=\int _{0}^{\infty }w^{-1-2\alpha +2\nu }\hat{f}_{\mu ,\alpha ,\beta ,\nu }^{\theta }(w)\overline{\hat{g}^\theta }_{\mu ,\alpha ,\beta ,\nu }(w) dw. \end{aligned}$$

(5)

Let $\varphi ,\psi \in L^1_{\mu ,\alpha ,\nu }(I)$. Then fractional Hankel-type integral convolution $(\varphi \#_\theta \psi )(t)$ and Hankel-type integral translation $(\tau ^\theta _{x} \varphi )(w)$ are respectively defined as:

$$\begin{aligned} (\varphi \#_\theta \psi )(t)=\nu \beta C^\theta _{\mu ,\alpha }\int ^{\infty }_0 \varphi (w)e^{\frac{i\beta }{2}w^{2\nu }\cot \theta } (\tau ^\theta _{t} \psi )(w) w^{\nu \mu -\alpha +2\nu -1}dw,\ \end{aligned}$$

(6)

where $0< t,w <\infty ,$ and

$$\begin{aligned} (\tau ^\theta _{t} \psi )(w)=\psi ^\theta (t,w)=\nu \beta C^\theta _{\mu ,\alpha }\int ^{\infty }_0 \varphi (z) D^\theta _{\mu ,\alpha ,\beta ,\nu }(t,w,z)e^{\frac{i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu -\alpha +2\nu -1}dz, \end{aligned}$$

(7)

provided that the above integrals exist, and being

$$\begin{aligned}&D^{\theta }_{\mu ,\alpha ,\beta ,\nu }(t,w,z) =\nu \beta (\csc \theta )^{2\alpha } C^{-\theta }_{\mu ,\alpha }\int _0^\infty (ts)^\alpha (ws)^\alpha (ts)^\alpha J_{\mu }\left( \beta (ts)^\nu \csc \theta \right) \nonumber \\&\quad \times J_{\mu }\left( \beta (ws)^\nu \csc \theta \right) J_{\mu }\left( \beta (zs)^\nu \csc \theta \right) e^{-\frac{i\beta }{2}(t^{2\nu }+w^{2\nu }+z^{2\mu })\cot \theta }s^{-1-3\alpha +2\nu -\nu \mu }ds.\nonumber \\ \end{aligned}$$

(8)

Moreover for $\nu \mu +2\nu -\alpha \ge 1$,

$$\begin{aligned}&\nu \beta C^\theta _{\mu ,\alpha }\int ^{\infty }_0z^{-1-2\alpha +2\nu }(zs)^{\alpha }J_{\mu }\left( \beta (zs)^\nu \csc \theta \right) e^{\frac{i\beta }{2}(z^{2\nu }+s^{2\nu })\cot \theta }\nonumber \\&\quad \times D^\theta _{\mu ,\alpha ,\beta ,\nu }(t,w,z)dz =(\csc \theta )^{2\alpha }(ts)^{\alpha }J_{\mu }\left( \beta (ts)^\nu \csc \theta )\right) (ws)^{\alpha }J_{\mu }\left( \beta (ws)^\nu \csc \theta \right) \nonumber \\&\quad \times e^{-\frac{i\beta }{2}(t^{2\nu }+w^{2\nu })\cot \theta }e^{\frac{i\beta }{2}s^{2\nu }\cot \theta }s^{-\alpha -\nu \mu }, \end{aligned}$$

(9)

and

$$\begin{aligned}&\nu \beta C^\theta _{\mu ,\alpha }\int ^{\infty }_0 z^{\nu \mu -\alpha +2\nu -1} D^\theta _{\mu ,\alpha ,\beta ,\nu }(t,w,z)e^{\frac{i\beta }{2}z^{2\nu }\cot \theta }dz\nonumber \\&\qquad =\frac{\beta ^{\mu }(tw)^{\alpha +\nu \mu }(\csc \theta )^{2\alpha +\mu }e^{-\frac{i\beta }{2}(t^{2\nu }+w^{2\nu })\cot \theta }}{2^{\mu }\Gamma (\mu +1)}. \end{aligned}$$

(10)

The mathematical theory of wave packet analysis on $\mathbb {R}$ is originated from dyadic dilations, integral translations and crips modulation of a particular signal [2, 3]. Theory of this paper depending on the ideas of fractional wavelets given by [4, 8, 9], the fractional Hankel-type integral wavelet $\psi _{b, a, \theta }$ as the dilation and translation of function $\psi \in L^2_{\mu ,\alpha , \nu }(I)$ with the parameters $a > 0$, $b \ge 0$ is mathematically defined as

$$\begin{aligned}&\psi _{b,a,\theta }(t)\nonumber \\&= D_a\left( \tau ^\theta _b\psi \right) (t)=D_a\psi ^\theta (b,t)=a^{-\frac{1}{2}-2\nu +2\alpha }e^{\frac{i\beta }{2}\left( \frac{1}{a^{2\nu }}-1\right) \left( t^{2\nu }+b^{2\nu }\right) \cot \theta }\psi ^\theta \left( \frac{b}{a},\frac{t}{a}\right) \nonumber \\&= a^{-\frac{1}{2}-2\nu +2\alpha }e^{\frac{i\beta }{2}\left( \frac{1}{a^{2\nu }}-1\right) \left( t^{2\nu }+b^{2\nu }\right) \cot \theta } \nu \beta C^\theta _{\mu ,\alpha } \int _0^\infty \psi (z)D^\theta _{\mu ,\alpha ,\beta ,\nu }\left( \frac{b}{a},\frac{t}{a},z\right) \nonumber \\&\quad \times e^{\frac{i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu -\alpha +2\nu -1}dz, \end{aligned}$$

(11)

where $D_a$ denotes the fractional dilation operator.

A fractional Hankel-type integral wavelet is a function $\psi \in L^2_{\mu ,\alpha ,\nu }(I)$, which satisfies the following condition:

$$\begin{aligned} C^{\theta ,\mu ,\alpha ,\beta ,\nu }_\psi =\int _0^\infty \frac{\vert \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu +\alpha }\psi (z)\right) (w)\vert ^2}{\nu ^2\beta ^2 w^{1+2\nu \mu +2\alpha }}dw<\infty , \end{aligned}$$

(12)

and it is known as the admissibility condition of the fractional Hankel-type integral wavelet.

Proposition 1

If$\psi \in L_{\mu ,\alpha ,\nu }^2(I)$, then the fractional powers of the Hankel-type integral transformation of$\psi _{b,a,\theta }$isgiven by

$$\begin{aligned} \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }\psi _{b,a,\theta }\right) (w)= & {} \frac{1}{\sqrt{a}}(aw)^{\nu \mu -\alpha }e^{-\frac{i\beta }{2}\left( (a^{2\nu }-1)w^{2\nu }+b^{2\nu }\right) \cot \theta }(bw\csc \theta )^\alpha \\&\times J_{\mu }\big (\beta (bw)^\nu \csc \theta \big ) \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu +\alpha }\psi (z)\right) (aw). \end{aligned}$$

Proof

See [7]. $\square$

Lemma 1

If$K^\theta (t, w)$is kernel of the transformation (2) then

$$\begin{aligned}&(a)\; \overline{K^{\theta }}(w, t)= K^{-\theta }(w, t),\\&(b)\; \int _0^\infty K^{\theta _1}(t, w)K^{\theta _2}(w, z) dw = K^{\theta _1 + \theta _2}(t, z), \\&(c)\; \int _0^\infty K^{\theta }(a, w)K^{-\theta }(w, t) dw = \delta (t-a). \end{aligned}$$

Proof

(a) Since for $\theta \ne n\pi$

$$\begin{aligned} K^{-\theta }(w,t)= & \, \nu \beta C_{\mu ,\alpha }^{-\theta } (tw)^{\alpha }J_\mu (-\beta (wt)^{\nu }\csc (-\theta ))e^{\frac{i\beta }{2}(t^{2\nu }+\omega ^{2\nu })\cot (-\theta )}w^{-1-2\alpha +2\nu } \\= & \, \nu \beta (-1)^{1+\mu }\frac{e^{-i(1+\mu )(\theta +\frac{\pi }{2})}}{\sin \theta } J_\mu (\beta (wt)^\nu \csc \theta )e^{-\frac{i\beta }{2}(t^{2\nu }+\omega ^{2\nu })\cot \theta }w^{-1-2\alpha +2\nu } \\= & \, \overline{\nu \beta \frac{e^{i(1+\mu )(\theta -\frac{\pi }{2})}}{\sin \theta } J_\mu (\beta (wt)^\nu \csc \theta )e^{\frac{i\beta }{2}(t^{2\nu }+\omega ^{2\nu })\cot \theta }w^{-1-2\alpha +2\nu }} \\= & \, \overline{K^\theta } (w,t). \end{aligned}$$

Similarly (b) and (c) can be proved easily by using the property (a). $\square$

2 Fractional Hankel-type integral wavelet packet transformation

As per [2, 3, 5, 8], the fractional Hankel-type integral wavelet packet transformation is defined as:

$$\begin{aligned} \left( W P^\theta _\psi f\right) (u, b, a)= \int _0^\infty K^\theta (t, u) \overline{\psi }_{b,a, \theta }(t) f(t) dt, \end{aligned}$$

(13)

where $\psi _{b,a, \theta }$ as fractional wavelet.

More precisely, the fractional Hankel-type integral wavelet packet transformation (FrHWPT) is given by

$$\begin{aligned}&\left( W P^\theta _\psi f\right) (u, b, a)= \nu \beta C^\theta _{\mu ,\alpha }\int _0^\infty (tu)^{\alpha } J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+u^{2\nu }) \cot \theta }\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \times ~~t^{-1-2\alpha +2\nu }\overline{\psi }_{b,a, \theta }(t) f(t) dt. \end{aligned}$$

(14)

Now assume that $f_u(t)= \nu \beta C^\theta _{\mu ,\alpha }(tu)^{\alpha } J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+u^{2\nu }) \cot \theta } f(t)$ and using Parseval’s identity in (14) we get

$$\begin{aligned} \left( W P^\theta _\psi f\right) (u, b, a)= \int _0^\infty \hat{f}_u^\theta (w)\overline{\hat{\psi }}_{b, a, \theta }(w) w^{-1-2\alpha +2\nu }dw. \end{aligned}$$

(15)

Now we see that

$$\begin{aligned} \hat{f}_u^\theta (w) & = \, \nu \beta C^\theta _{\mu ,\alpha }\int _0^\infty (tu)^{\alpha } J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+u^{2\nu }) \cot \theta } f_u(t) t^{-1-2\alpha +2\nu } dt\\ & = \, \nu ^2 \beta ^2 (C^\theta _{\mu , \alpha })^2 \int _0^\infty (tw)^\alpha J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+u^{2\nu }) \cot \theta }\\&\quad \times\, \left( (tu)^{\alpha } J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+u^{2\nu }) \cot \theta } f(t)\right) t^{-1-2\alpha +2\nu } dt. \end{aligned}$$

Using (9), (2) and (7), we get

$$\begin{aligned}&\hat{f}_u^\theta (w)=\nu \beta C^\theta _{\mu ,\alpha } (\csc \theta )^{-2\alpha }e^{i\beta (w^{2\nu } + u^{2\nu })\cot \theta }\nonumber \\&\quad \quad \quad \times \tau ^\theta _{u}\left( z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta } H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }f(t)\big )(z)\right) (w). \end{aligned}$$

(16)

Therefore

$$\begin{aligned}&\left( W P^\theta _\psi f\right) (u, b, a)\nonumber \\&=\frac{\nu \beta C^\theta _{\mu , \alpha }e^{i \beta u^{2\nu }\cot \theta }}{(\csc \theta )^{2\alpha }\sqrt{a}} \int _0^\infty \tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }f(t)\big )(z)\Big )(w)\nonumber \\&\quad \times (aw)^{-\nu \mu -\alpha } e^{\frac{i \beta }{2}a^{2\nu }w^{2\nu }}e^{\frac{i\beta }{2}(w^{2\nu }+ b^{2\nu })\cot \theta }(bw)^\alpha J_\mu \big (\beta (bw)^\nu \csc \theta \big )\nonumber \\&\quad \times \overline{ \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu +\alpha }\psi (z)\right) (aw)}w^{-1-2\alpha + 2\nu } dw. \end{aligned}$$

(17)

Remark 1

From (17) we can express the fractional Hankel- type integral wavelet packet transformation $\left( W P^\theta _\psi f\right) (u, b, a)$ of a function $f \in L^2_{\mu , \alpha , \nu }(I)$ as:

$$\begin{aligned}&\left( W P^\theta _\psi f\right) (u, b, a)\\&=\frac{e^{i \beta u^{2\nu }\cot \theta }}{(\csc \theta )^{2\alpha }\sqrt{a}} H^{\theta }_{\mu , \alpha , \beta , \nu }\Big [\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }f(t)\big )(z)\Big )\\&\quad (w)\Big ] (aw)^{-\mu \nu - \alpha } e^{\frac{i\beta }{2}a^{2\nu }w^{2\nu }\cot \theta } \overline{ \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }z^{\nu \mu +\alpha }\psi (z)\right) (aw)}\Big ] (b). \end{aligned}$$

Theorem 1

If$\psi _1$ and $\psi _2$are two fractional Hanlel- type integral wavelets,$f_1$, $f_2$$\in L^2_{\mu , \alpha , \nu }(I)$and forscalars$\alpha _1$ and $\alpha _2$we have

$$\begin{aligned}&(a) \left( W P^\theta _{\psi _1} (\alpha _1f_1 + \alpha _2 f_2)\right) (u, b, a) = \alpha _1\left( W P^\theta _{\psi _1} f_1\right) (u, b, a) + \alpha _2 \left( W P^\theta _{\psi _2} f_2\right) (u, b, a),\\&(b) \left( W P^\theta _{\alpha _1\psi _1 + \alpha _2 \psi _2} f_1\right) (u, b, a) = \overline{\alpha _1} \left( W P^\theta _{\psi _1} f_1\right) (u, b, a) + \overline{\alpha _2} \left( W P^\theta _{\psi _2} f_1\right) (u, b, a). \end{aligned}$$

Proof

(a) Using (13), we have

$$\begin{aligned}&\left( W P^\theta _{\psi _1} (\alpha _1f_1 + \alpha _2 f_2)\right) (u, b, a) \\&\quad= \int _0^\infty K^{\theta }(t,u)\overline{\psi _1}_{b,a,\theta }(t)(\alpha _1f_1 + \alpha _2 f_2)(t) dt \\&\quad= \alpha _1 \int _0^\infty K^{\theta }(t,u)\overline{\psi _1}_{b,a,\theta }(t)f_1(t)dt + \alpha _2 \int _0^\infty K^{\theta }(t,u)\overline{\psi _1}_{b,a,\theta }(t)f_2(t)dt \\&\quad= \alpha _1\left( W P^\theta _{\psi _1} f_1\right) (u, b, a) + \alpha _2 \left( W P^\theta _{\psi _2} f_2\right) (u, b, a). \end{aligned}$$

Similarly (b) can be proved as (a). $\square$

Theorem 2

If$\psi _1$ and $\psi _2$are two fractional wavelets and$(W P^\theta _{\psi _1} f)(u, b, a)$and$(W P^\theta _{\psi _2} g)(u, b, a)$denote the fractional Hankel-type integral wavelet packet transformations of the functionsfandgrespectively, then

$$\begin{aligned}&\int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a) \overline{(W P^\theta _{\psi _2} g)(u, b, a)} b^{-1-2\alpha +2\nu } db da \nonumber \\&\quad \quad \quad \quad = (\csc \theta )^{4\alpha } C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu } \int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 f(x)\overline{g(x)} x^{-1-2\alpha + 2\nu }dx, \end{aligned}$$

where

$$\begin{aligned}&C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu } = (\nu \beta )^{-2} \int _0^\infty w^{-1-2\nu \mu -2\alpha } \left( H^\theta _{\mu ,\alpha ,\beta ,\nu }\left( z^{\nu \mu +\alpha }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }\psi _1(z)\right) \right) (w)\\&\quad \quad \quad \quad \quad \times \overline{\left( H^\theta _{\mu ,\alpha ,\beta ,\nu }\left( z^{\nu \mu +\alpha }e^{\frac{-i\beta }{2}z^{2\nu }\cot \theta }\psi _2(z)\right) \right) (w)} dw. \end{aligned}$$

Proof

Using Remark 1 and (5), we get

$$\begin{aligned}&\int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a) \overline{(W P^\theta _{\psi _2} g)(u, b, a)} b^{-1-2\alpha +2\nu } db da \nonumber \\&= \frac{\nu ^2 \beta ^2 C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu }}{(\csc \theta )^{-4\alpha }} \int _0^\infty \tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta } f(t)\big )(z)\Big )(w)\nonumber \\&\quad \times \overline{\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }g(t)\big )(z)\Big )(w)} w^{-1-2\alpha +2\nu } dw\nonumber \\&= \frac{\nu ^2 \beta ^2 C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu }}{(\csc \theta )^{-4\alpha }} \int _0^\infty H^\theta _{\mu , \alpha , \beta , \nu }\Big [\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta } \nonumber \\&\quad \times f(t)\big )(z)\Big )(w)\Big ]~~\overline{ H^\theta _{\mu , \alpha , \beta , \nu }\Big [\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta } H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }} \nonumber \\&\quad \times \overline{g(t)\big )(z)\Big )(w)\Big ]}~~ x^{-1-2\alpha +2\nu } dx. \end{aligned}$$

(18)

Now we see that

$$\begin{aligned}&H^\theta _{\mu , \alpha , \beta , \nu }\Big [\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }f(t)\big )(z)\Big )(w)\Big ](x)\nonumber \\&= \nu \beta \int _0^\infty (xw)^{\alpha } J_\mu \big (\beta (xw)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(x^{2\nu }+w^{2\nu }) \cot \theta }w^{-1-2\alpha +2\nu } \Big ( \nu \beta C^\theta _{\mu ,\alpha }\nonumber \\&\quad \times \int _0^\infty D^\theta _{\mu , \alpha , \beta , \nu }(u, w, z)H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }f(t)\big )(z) z^{-1-2\alpha +2\nu }dz \Big ) dw\nonumber \\&= (\csc \theta )^{2\alpha }e^{2i(1+ \mu )(\theta - \frac{\pi }{2})}e^{-\frac{i\beta }{2}u^{2\nu }\cot \theta }e^{\frac{3i}{2}\beta x^{2\nu }\cot \theta }(u x)^\alpha J_\mu \big (\beta (ux)^\nu \csc \theta \big )f(x).\nonumber \\ \end{aligned}$$

(19)

Similarly proceeding as the above we get

$$\begin{aligned}&\overline{H^\theta _{\mu , \alpha , \beta , \nu }\Big [\tau ^\theta _{u}\Big (z^{-\nu \mu -\alpha } e^{-\frac{i\beta }{2}z^{2\nu } \cot \theta }H^\theta _{\mu , \alpha , \beta , \nu }\big (t^{\nu \mu + \alpha }e^{\frac{i\beta }{2}t^{2\nu }\cot \theta }g(t)\big )(z)\Big )(w)\Big ](x)}\nonumber \\&= (\csc \theta )^{2\alpha }e^{-2i(1+ \mu )(\theta - \frac{\pi }{2})}e^{\frac{i\beta }{2}u^{2\nu }\cot \theta }e^{-\frac{3i}{2}\beta x^{2\nu }\cot \theta }(u x)^\alpha J_\mu \big (\beta (ux)^\nu \csc \theta \big )\overline{g(x)}.\nonumber \\ \end{aligned}$$

(20)

Using (19) and (20) in (18) we get

$$\begin{aligned}&\int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a) \overline{(W P^\theta _{\psi _2} g)(u, b, a)} b^{-1-2\alpha +2\nu } db da \nonumber \\&\quad \quad \quad = (\csc \theta )^{4\alpha } C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu } \int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 f(x)\overline{g(x)} x^{-1-2\alpha + 2\nu }dx. \end{aligned}$$

$\square$

Remark 2

The following are deductions of Theorem 2:

• (i) If $f=g$, then

  $$\begin{aligned}&\int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a) \overline{(W P^\theta _{\psi _2} f)(u, b, a)} b^{-1-2\alpha +2\nu } db da \nonumber \\&\quad \quad \quad = (\csc \theta )^{4\alpha } C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu } \int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 |f(x)|^2 x^{-1-2\alpha + 2\nu }dx. \end{aligned}$$

• (ii) If $\psi _1=\psi _2=\psi$, then

  $$\begin{aligned}&\int _0^\infty \int _0^\infty (W P^\theta _{\psi } f)(u, b, a) \overline{(W P^\theta _{\psi } g)(u, b, a)} b^{-1-2\alpha +2\nu } db da \nonumber \\&\quad \quad \quad = (\csc \theta )^{4\alpha } C_{\psi }^{\theta , \mu , \alpha , \beta , \nu } \int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 f(x)\overline{g(x)} x^{-1-2\alpha + 2\nu }dx. \end{aligned}$$

• (iii) If $f=g$ and $\psi _1=\psi _2=\psi$, then

  $$\begin{aligned}&\int _0^\infty \int _0^\infty |\left( W^\theta _{\psi } f\right) (u, b,a)|^2b^{-1-2\alpha +2\nu }dbda\\&\quad \quad \quad =(\csc \theta )^{4\alpha } C_{\psi }^{\theta , \mu , \alpha , \beta , \nu } \int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 |f(x)|^2 x^{-1-2\alpha + 2\nu }dx \end{aligned}$$

  where $C^{\theta ,\mu ,\alpha ,\beta ,\nu }_{\psi }$ is given by (11).

Theorem 3

(Inversion Formula) Let$f\in L^2_{\mu , \alpha , \nu }(I)$. Then f can be reconstructed involving the fractional Hankel-type transformation by the formula

$$\begin{aligned} f(t) &= \frac{\nu \beta \overline{C^\theta _{\mu , \alpha }}e^{-\frac{i\beta }{2}(u^{2\nu }+ t^{2\nu })\cot \theta }}{(\csc \theta )^{4 \alpha } C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu }(u t)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )} \int _0^\infty \int _0^\infty \left( W^\theta _{\psi } f\right) (u, b,a)\nonumber \\& \quad \times (\psi _2)_{b, a, \theta }(t)b^{-1-2\alpha +2\nu }dbda \end{aligned}$$

where$C^{\theta , \mu , \alpha , \beta , \nu }_{\psi _1, \psi _2}$is as in Theorem2.

Proof

Form Theorem 2, we have

$$\begin{aligned}&(\csc \theta )^{4\alpha }C^{\theta , \mu , \alpha , \beta , \nu }_{\psi _1, \psi _2}\int _0^\infty \Big ((ut)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )\Big )^2 f(t)\overline{g(t)}t^{-1-2\alpha +2\nu } dt \\&= \int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a) \overline{(W P^\theta _{\psi _2} g)(u, b, a)} b^{-1-2\alpha +2\nu } db da\\&= \int _0^\infty \Big ( \nu \beta C^\theta _{\mu ,\alpha }\int _0^\infty \int _0^\infty (W P^\theta _{\psi _1} f)(u, b, a)(tw)^{\alpha } J_\mu \big (\beta (tu)^\nu \csc \theta \big )e^{\frac{i\beta }{2}(t^{2\nu }+w^{2\nu }) \cot \theta }\\&\quad \times (\psi _2)_{b, a, \theta } b^{-1-2\alpha +2\nu } db da\Big )\overline{g(t)}t^{-1-2\alpha +2\nu } dt. \end{aligned}$$

On equating we get

$$\begin{aligned} f(t)= & {} \frac{\nu \beta \overline{C^\theta _{\mu , \alpha }}e^{-\frac{i\beta }{2}(u^{2\nu }+ t^{2\nu })\cot \theta }}{(\csc \theta )^{4 \alpha } C_{\psi _1, \psi _2}^{\theta , \mu , \alpha , \beta , \nu }(u t)^\alpha J_\mu \big (\beta (ut)^\nu \csc \theta \big )} \int _0^\infty \int _0^\infty \left( W^\theta _{\psi } f\right) (u, b,a)\nonumber \\&\quad \times (\psi _2)_{b, a, \theta }(t)b^{-1-2\alpha +2\nu }dbda. \end{aligned}$$

(21)

$\square$

3 Some examples

Example 1

Assume that $f(t)= \frac{\delta (t-c)}{t^{2\nu -1}}, 0< c < \infty$. Then the fractional Hankel-type integral wavelet packet transformation of f(t) is given by

$$\begin{aligned}&\left( W^\theta _{\psi } f\right) (u, b,a)\\&= \int _0^\infty K^\theta (t, u) \overline{\psi }_{b,a, \theta }(t) f(t) dt\\&= \nu ^3\beta ^3 C^\theta _{\mu , \alpha }(\csc \theta )^{2 + 2\alpha } a^{-\frac{1}{2}- 2\nu } (bu)^{\alpha }e^{\frac{i\beta }{2}(u^{2\nu }+ b^{2\nu })\cot \theta }\int _0^\infty \overline{\psi (z)} z^{\nu \mu +2\nu -1}dz \\&\quad \times \int _0^\infty x^{-1 + 2\nu -\nu \mu }J_\mu \left( \beta \Big (\frac{b}{a}x\Big )^\nu \csc \theta \right) J_\mu \left( \beta \big (xz\big )^\nu \csc \theta \right) dx\\&\quad \times \left( \int _0^\infty J_\mu \left( \beta \Big (\frac{t}{a}x\Big )^\nu \csc \theta \right) J_\mu \left( \beta \big (tu\big )^\nu \csc \theta \right) e^{i\beta t^{2\nu }\cot \theta }\delta (t - c) dt\right) \\&= \nu ^3\beta ^3 C^\theta _{\mu , \alpha }(\csc \theta )^{2 + 2\alpha } a^{-\frac{1}{2}- 2\nu } (bu)^{\alpha }e^{\frac{i\beta }{2}(u^{2\nu }+ b^{2\nu }+ 2c^{2\nu })\cot \theta }J_\mu \left( \beta \big (tu\big )^\nu \csc \theta \right) \\&\quad \times \int _0^\infty \overline{\psi (z)} z^{\nu \mu +2\nu -1}dz \int _0^\infty x^{-1 + 2\nu -\nu \mu }J_\mu \left( \beta \Big (\frac{b}{a}x\Big )^\nu \csc \theta \right) \\&\quad \times J_\mu \left( \beta \big (xz\big )^\nu \csc \theta \right) J_\mu \left( \beta \Big (\frac{c}{a}x\Big )^\nu \csc \theta \right) dx. \end{aligned}$$

Using [10, p.41], we get

$$\begin{aligned}&\left( W^\theta _{\psi } f\right) (u, b,a)\\&= \frac{\nu ^2 \beta ^{3-3\mu } C^\theta _{\mu ,\alpha }}{\Gamma (\mu + \frac{1}{2}) \Gamma \frac{1}{2}} (\csc \theta )^{2+2\alpha -3\mu } u^\alpha b^{\alpha -\nu }c^{-\nu }2^{\mu -1} a^{-\frac{1}{2}}e^{\frac{i\beta }{2}(u^{2\nu }+ b^{2\nu }+ 2c^{2\nu })\cot \theta }\\&\quad \times J_\mu \left( \beta \big (uc\big )^\nu \csc \theta \right) \int _0^\infty \overline{\psi (z)} z^{2\nu -1}\Delta ^{2\mu -1}dz, \end{aligned}$$

where $\triangle$ denotes the area of a triangle having sides $\Big (\beta \left( \frac{b}{a}\right) \csc \theta \Big ), \Big (\beta \left( \frac{c}{a}\right) \csc \theta \Big ), \left( \beta z^\nu \csc \theta \right)$ such a triangle exists.

Example 2

Assume $f(t)= e^{-i t \cot \theta }$, then

$$\begin{aligned}&\left( W^\theta _{\psi } f\right) (u, b,a)\nonumber \\&= \nu ^3\beta ^3 C^\theta _{\mu , \alpha }(\csc \theta )^{2 + 2\alpha } a^{-\frac{1}{2}- 2\nu } (bu)^{\alpha }e^{\frac{i\beta }{2}(u^{2\nu }+ b^{2\nu })\cot \theta }\int _0^\infty \overline{\psi (z)} z^{\nu \mu +2\nu -1}dz\nonumber \\&\quad \times \int _0^\infty x^{-1 + 2\nu -\nu \mu }J_\mu \left( \beta \Big (\frac{b}{a}x\Big )^\nu \csc \theta \right) J_\mu \left( \beta \big (xz\big )^\nu \csc \theta \right) dx\nonumber \\&\quad \times \left( \int _0^\infty t^{2\nu -1} J_\mu \left( \beta \Big (\frac{t}{a}x\Big )^\nu \csc \theta \right) J_\mu \left( \beta \big (tu\big )^\nu \csc \theta \right) dt\right) . \end{aligned}$$

(22)

Using [1, p.6] and (4) in (22), we get

$$\begin{aligned}&\left( W^\theta _{\psi } f\right) (u, b,a)\nonumber \\&= \nu ^2\beta ^3 C^\theta _{\mu , \alpha }(\csc \theta )^{2\alpha +2} a^{-\frac{1}{2}-2\nu } (bu)^{\alpha }e^{\frac{i\beta }{2}(u^{2\nu }+ b^{2\nu })\cot \theta }\int _0^\infty \overline{\psi (z)} z^{\nu \mu +2\nu -1}dz\nonumber \\&\quad \times \int _0^\infty x^{-\nu \mu +2\nu -1}J_\mu \left( \beta \Big (\frac{b}{a}x\Big )^\nu \csc \theta \right) J_\mu \left( \beta \big (xz\big )^\nu \csc \theta \right) \frac{a^\nu }{\beta x^\nu \csc \theta }\\&\quad \times \delta \Big ( \frac{\beta x^\nu \csc \theta }{a^\nu } - \beta u^\nu \csc \theta \Big )dx\\&= (\csc \theta )^{2\alpha }u^{-\nu \mu + 2\alpha -2\nu +1}a^{-\nu \mu -\frac{1}{2}}K^\theta (u, b)\int _0^\infty \overline{\psi (z)}z^{\nu \mu + 2\nu - 1}J_\mu \left( \beta \big (auz\big )^\nu \csc \theta \right) dz. \end{aligned}$$