The ${\mathcal{T}}_{\mathbf{e}}$ Transform: A High-Resolution Integral Transform and Its Key Properties

Abstract

In this paper, we present six new contributions: two novel definitions and four groundbreaking theorems related to the theoretical foundations of the integral ${\mathcal{T}}_e$ transform, with a specific focus on analyzing functions with integrable modulus. The definitions referred to the ${\mathcal{T}}_e$ window and the ${\mathcal{T}}_e$ transform in two parameters, respectively. The theorems provide the main theoretical basis for the ${\mathcal{T}}_e$ transform: the existence of the ${\mathcal{T}}_e$ transform in two parameters, the ${\mathcal{T}}_e$ transform $\in {\mathrm{L}}^1\left(\mathbb{R}\right)$, the existence of the inverse ${\mathcal{T}}_e$ transform, and uniqueness of the ${\mathcal{T}}_e$ transform. These results reveal the importance of the fact that the ${\mathcal{T}}_e$ transform only depends on two parameters (translation and dyadic frequency), obtaining its inverse transformation more directly; hence, breaking through a new approach in function analysis by representing a function in the scale-frequency plane. The theoretical results presented in this paper are supported by the previous works of the authors.

1. Introduction

This paper presents six new contributions to the integral ${\mathcal{T}}_e$ transform in the field of mathematics applied to engineering.

The first contribution focuses on defining the ${\mathcal{T}}_e$ window, while the second contribution introduces the definition of the ${\mathcal{T}}_e$ transform in two parameters. The third contribution presents a theorem regarding the existence of the ${\mathcal{T}}_e$ transform, while the fourth establishes this transform belongs to $\in {\mathrm{L}}^1\left(\mathbb{R}\right)$. Furthermore, the fifth contribution demonstrates the existence of the inverse ${\mathcal{T}}_e$ transform, and lastly, the sixth contribution provides a uniqueness theorem for the ${\mathcal{T}}_e$ transform. The theoretical results presented in this paper were validated by the authors of the works shown in [1,2,3,4].

The problem of function recovery ($f\left(t\right),\forall t\in \mathbb{R}$) is very important in the field of applied mathematics [5,6,7]. In [2], the authors introduce the ${\mathcal{T}}_e$ transform, see Equation (1), as a dependent map of three parameters. This transform has as its main advantages the attenuation of the cross-terms and the isolation of the frequency components [2], which are achieved through the dyadic frequency spectrum. To have an integral transform dependent on more than two parameters makes it difficult to represent them and obtaining their inverse integral transform becomes very difficult.

$${\mathcal{T}}_e≔{\widehat{f}}_{DTe}\left[f\left(t\right)\right]\left(\mu ,\xi ,\vartheta \right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(t\right)w_{\mu }\left(t\right){\psi }_{\xi ,\mu }\left(t\right)e^{-i2\pi \vartheta t}dt,$$

(1)

where:

• $\vartheta$ is the dyadic linear frequency [2],
• $t$ represents time or space, and
• $w_{\mu }\left(t\right)=w\left(t-\mu \right),\forall \mu \in \mathbb{R}$ and ${\psi }_{\xi ,\mu }\left(t\right)=\frac{1}{\sqrt{2^{\xi }}}\psi \left(\frac{t}{2^{\xi }}-\mu \right),\forall \xi \in \mathbb{Z}$ are the window function and the dyadic wavelet function, respectively.

While [2] presents a procedure for recovering a function $f\left(\upsilon \right)$,$\forall \upsilon \in \mathbb{R}$, where $\upsilon$ represents either time or space, from its ${\mathcal{T}}_e$ transform, there is currently no theorem that provides a generalized equation for the recovery of $f\left(\upsilon \right)$.

Another challenge highlighted in Equation (1) is the presence of a function dependent on three parameters $\mu ,\xi ,\vartheta$. This complexity implies that obtaining its inverse integral transform may not be a straightforward process. Furthermore, additional issues arise, such as the lack of clarity regarding the existence and uniqueness of the ${\mathcal{T}}_e$ transform, as well as the functional space in which the ${\mathcal{T}}_e$ transform is defined. Therefore, this manuscript addresses these existing gaps.

The paper is organized as follows. Section 2 contains the theoretical background used in the proposed definitions and theorems. Section 3 shows the definitions and proposed theorems with their proofs. Section 4 shows the transformed pairs calculated by the ${\mathcal{T}}_e$ transform. Finally, the main conclusions are shown in Section 5.

2. Theoretical Background

This section provides the theoretical foundations used for analyzing a function $f\left(\mathsf{\upsilon }\right)$ belonging to the subspace ${\mathrm{L}}^1\left(\mathbb{R}\right)\cap {\mathrm{L}}^2\left(\mathbb{R}\right)$, which is dense in ${\mathrm{L}}^2\left(\mathbb{R}\right)$. These theoretical foundations will be employed in Section 3.

Fourier Transform and Its Inverse

The Fourier transform for a function $f\left(\mathsf{\upsilon }\right)$ that is absolutely integrable is defined in Equation (2) and its inverse is defined in Equation (3) according to [8,9].

$$F\left(\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right)e^{-i\mathsf{\omega }\mathsf{\upsilon }}d\mathsf{\upsilon },$$

(2)

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F\left(\omega \right)e^{i\omega \upsilon }d\omega ,$$

(3)

where:

• $\omega$ is the angular frequency,
• $\mathsf{\upsilon }$ represents time or space, and,
• $i$ is the complex unit.

• Parseval’s Theorem [10,11,12], Equation (4), and Plancherel’s Theorem [13,14,15] Equation (5), respectively

If $f\left(\mathsf{\upsilon }\right)$ and $h\left(\mathsf{\upsilon }\right)$ are two functions on ${\mathrm{L}}^1\left(\mathbb{R}\right)\cap {\mathrm{L}}^2\left(\mathbb{R}\right)$, then

$$\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right)h^{\ast }\left(\mathsf{\upsilon }\right)dt=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F\left(\omega \right)H^{\ast }\left(\omega \right)d\omega .$$

(4)

For $f\left(\mathsf{\upsilon }\right)=h\left(\mathsf{\upsilon }\right)$ it is obtained that

$$\underset{-\infty }{\overset{\infty }{\int }}{\left|f\left(\mathsf{\upsilon }\right)\right|}^{2}dt=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{\left|F\left(\omega \right)\right|}^{2}d\omega ,$$

(5)

where $F\left(\omega \right)$ is the Fourier transform $f\left(\mathsf{\upsilon }\right)$, and $H^{\ast }\left(\omega \right)$ is the complex conjugate of the Fourier transform of $h\left(\mathsf{\upsilon }\right)$.

• Fubini’s Theorem [9,16,17]

If ${\int }_{-\infty }^{\infty }\left({\int }_{-\infty }^{\infty }\left|f({\mathsf{\upsilon }}_1,{\mathsf{\upsilon }}_2)\right|d{\mathsf{\upsilon }}_1\right)d{\mathsf{\upsilon }}_2<+\infty$, then Equation (6) holds.

$$\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}f\left({\mathsf{\upsilon }}_1,{\mathsf{\upsilon }}_2\right)d{\mathsf{\upsilon }}_{1}d{\mathsf{\upsilon }}_2=\underset{-\infty }{\overset{\infty }{\int }}\left(\underset{-\infty }{\overset{\infty }{\int }}f\left({\mathsf{\upsilon }}_1,{\mathsf{\upsilon }}_2\right)d{\mathsf{\upsilon }}_1\right)d{\mathsf{\upsilon }}_2=\underset{-\infty }{\overset{\infty }{\int }}\left(\underset{-\infty }{\overset{\infty }{\int }}f\left({\mathsf{\upsilon }}_1,{\mathsf{\upsilon }}_2\right)d{\mathsf{\upsilon }}_2\right)d{\mathsf{\upsilon }}_1$$

(6)

where $f\left({\mathsf{\upsilon }}_1,{\mathsf{\upsilon }}_2\right)$ represents an absolutely integrable function, permitting the interchangeability of the order of integration.

• Lebesgue’s Dominated Convergence Theorem [18,19]

Let $\left\{f_n\right\}$ be a sequence of measurable functions on $\mathrm{E}$ that converges punctually to the function $f\left(\mathsf{\upsilon }\right)$ and suppose that there exists an integrable function $\mathrm{G}$ on $\mathrm{E}$ such that $\left|f_n\right|\le \mathrm{G},\forall n\in N$, then

(a)

$f\left(\mathsf{\upsilon }\right)$ is integrable on $\mathrm{E}$,

(b)

${\int }_{E}f(\mathsf{\upsilon })d\mathsf{\upsilon }=\underset{n\to \infty }{\lim }{\int }_E{f}_n\left(\mathsf{\upsilon }\right)d\mathsf{\upsilon }$

• Function Composition Theorem [20,21]

Let $f\left(\mathsf{\upsilon }\right)$ be an integrable function on the interval $\left[\mathsf{A},\mathrm{B}\right]$, and let $g\left(\mathsf{\upsilon }\right)$ be a continuous function on the interval $\left[\mathsf{{\rm E}},\mathsf{{\rm Z}}\right]$. Suppose that $f\left(\left[\mathsf{A},\mathrm{B}\right]\right)\subset g\left(\left[\mathsf{{\rm E}},\mathsf{{\rm Z}}\right]\right)$. Then, $\left(g\circ f\right)\left(\mathsf{\upsilon }\right)$ is integrable on $\left[\mathsf{A},\mathrm{B}\right]$ $\forall \mathrm{A},\mathrm{B},\mathrm{E},\mathrm{Z}\in \mathbb{R}$.

3. Results

This section presents the definitions, deduction, and theorems related to the ${\mathcal{T}}_e$ transform.

Definition 1

(${\mathcal{T}}_{\mathrm{e}}$ Window). A bounded function ${\beta }_{\mu ,\xi }\left(\upsilon \right)$, centered at $\upsilon =0$, even, and in $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$ is a ${\mathcal{T}}_e$ window; if it is the product of a bounded function $w_{\mu }\left(\upsilon \right)$ with $‖w_{\mu }‖=1$ and a dyadic wavelet function, ${\psi }_{\mu ,\xi }\left(\upsilon \right)$, where $\mu$ and $\xi$ are the translation and scale parameters, respectively. Equation (7) shows the ${\mathcal{T}}_e$ window.

$${\beta }_{\mu ,\xi }\left(\upsilon \right)=w_{\mu }\left(\upsilon \right)\cdot {\psi }_{\mu ,\xi }\left(\upsilon \right),$$

(7)

where:

• ${\psi }_{\mu ,\xi }\left(\upsilon \right)=\frac{1}{\sqrt{2^{\xi }}}\psi \left(\frac{\upsilon }{2^{\xi }}-\mu \right)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$, and
• $w_{\mu }\left(\upsilon \right)=w\left(\upsilon -\mu \right)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$.

In applied mathematics, it is necessary that the ${\mathcal{T}}_e$ window belongs to $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$. This is because $L^2\left(\mathbb{R}\right)$ is not a subspace of $L^1\left(\mathbb{R}\right)$ [22]. Moreover, if ${\beta }_{\mu ,\xi }\left(\upsilon \right)\in L^2\left(\mathbb{R}\right)$ but ${\beta }_{\mu ,\xi }\left(\upsilon \right)\notin L^1\left(\mathbb{R}\right),$ then the Fourier transform of ${\beta }_{\mu ,\xi }\left(\upsilon \right)$ cannot be computed by Equation (2) because ${\beta }_{\mu ,\xi }\left(\upsilon \right)e^{-i\omega \upsilon }$ is not integrable [9], as per Lebesgue’s theorem. This implies the need to bound our universe of functions to the set $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$ dense in $L^2\left(\mathbb{R}\right)$.

The significance of combining two window functions, as seen in Equation (7), one without the scale parameter, $w_{\mu }\left(\upsilon \right)$, and another with the scale parameter, ${\psi }_{\mu ,\xi }\left(\upsilon \right)$, allows the possibility of discriminating spurious frequency components. The main goal of using two windows with mentioned features is to find a plugin that maximizes the attenuation of unwanted frequency features.

Recently, the authors of [3,4] have utilized the ${\mathcal{T}}_e$ transform as a basis in hydraulic and seismic applications. They reported as the main innovation the ${\mathcal{T}}_e$ cross-spectral density, along with ${\mathcal{T}}_e$ coherence and the ${\mathcal{T}}_e$ gram, respectively. In their work, they achieved favorable results using a Daubechies $45$ wavelet and a Kaiser-$6$ window. It is worth noting that the behavior of these windows is influenced by the parameter $\xi$, which, in turn, impacts the function ${\beta }_{\mu ,\xi }\left(\upsilon \right)$. Consequently, the function ${\beta }_{\mu ,\xi }\left(\upsilon \right)$ is also dependent on $\xi$.

Figure 1 shows an example for the previous discussion using a generic window function and a Daubechies $4$ wavelet [23,24,25]. The effect explained earlier gives a redundancy effect that offers the possibility for maximizing the noise attenuation and cross-terms attenuation. Note that in Figure 1a–d, the gray region represents the area removed after multiplication. The results of the ${\mathcal{T}}_e$ window after contracting the dyadic wavelet function or the window function favor noise attenuation. This effect increases the sensitivity, obtaining a multi-sensitive spectrum.

Deduction 1

(${\mathcal{T}}_{\mathrm{e}}$ Transform in Two Parameters). We can rewrite Equation (1) as shown in Equation (8), considering the parameter of shift-invariance $\mu =0$ [2], and at the scale ${\xi }_1$.

$${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\omega \right)=\frac{1}{\sqrt{2^{{\xi }_1}}}\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right){\beta }_{0,{\xi }_1}\left(\upsilon \right)e^{-i\omega \upsilon }d\upsilon ,$$

(8)

$${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\omega \right)=\frac{1}{\sqrt{2^{{\xi }_1}}}\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right){\beta }_0\left(\frac{\upsilon }{2^{{\xi }_1}}\right)e^{-i\omega \upsilon }d\upsilon .$$

(9)

If we make a change of parameters, in Equation (9),$\frac{\upsilon }{2^{{\xi }_1}}=\tau$ obtaining that $d\upsilon =2^{{\xi }_1}d\tau$, we realize that ${\beta }_0\left(\frac{\upsilon }{2^{\xi }}\right)$ is bounded and is multiplying the function $f\left(\upsilon \right)$. Then, by Figure 2, we rewrite Equation (9) as shown in Equation (10).

$${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(2^{{\xi }_1}\omega \right)=\frac{1}{\sqrt{2^{{\xi }_1}}}\underset{a}{\overset{b}{\int }}h\left(\tau \right)e^{-i\omega 2^{{\xi }_1}\tau }{2}^{{\xi }_1}d\tau .$$

(10)

If we compare Equation (10) with Equation (2), we can observe that we are in the presence of the Fourier transform of $h\left(\tau \right)$. After rearranging Equation (10), we obtain Equation (11).

$${\mathcal{T}}_e\left[h\left(\tau \right)\right]\left(2^{{\xi }_1}\omega \right)=\sqrt{2^{{\xi }_1}}H\left(2^{{\xi }_1}\omega \right).$$

(11)

This result indicates how the frequency domain is scaled. If we consider the translation parameter, then we can express the result of Equation (11) as ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)$ due to the shift-invariance property [2]. This indicates the possibility of expressing Equation (1) in two parameters, making way to Definition 2.

Expressing the ${\mathcal{T}}_e$ transform in two parameters allows us to interpret the two-dimensional behavior of the ${\mathcal{T}}_e$ spectrum for a function $f\left(\upsilon \right)$. From the point of view of applied mathematics, the ${\mathcal{T}}_e$ transform continues to provide localized information in dyadic time-frequency. Only when compacting its output information is it observed how the frequency is scaled by a dyadic factor. Previous research conducted by the authors of this paper demonstrates that such scaling enables the isolation of frequency components, the attenuation of cross-terms, and the identification of system behaviors [1,2,3,4].

Theorem 1

(Existence of the ${\mathcal{T}}_e$ Transform in Two Parameters). Let Equation (12) be the ${\mathcal{T}}_e$ transform for a function $f\left(\upsilon \right)$, it will be shown below that there exists the ${\mathcal{T}}_e$ transform in two parameters.

$${\mathcal{T}}_e\left[f\left(\upsilon \right)\right](\mu ,2^{\xi }\omega )=\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{-i\omega \upsilon }d\upsilon .$$

(12)

If $f\left(\upsilon \right)$ and ${\beta }_{\mu ,\xi }\left(\upsilon \right)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right),$ then $\exists {\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)$.

Proof. 

To carry out the demonstration, we will calculate the integral of the module of the integrand of Equation (12). This is

$$\underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}\right|d\mathsf{\upsilon }=\underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\right|\left|e^{-i\omega \mathsf{\upsilon }}\right|d\mathsf{\upsilon }.$$

Since $f\left(\upsilon \right)$ and ${\beta }_{\mu ,\xi }\left(\upsilon \right)$ are in $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right),$ then $f\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)$ is in $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$ which means $\left|f\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)\right|$ is absolutely integrable. Also, $\left|e^{-i\omega \upsilon }\right|=1$. Then, Equation (13) is fulfilled proving that the ${\mathcal{T}}_e$ transform of $f\left(\upsilon \right)$ exists.

$$\underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right)\right|\left|{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\right|\left|e^{-i\omega \mathsf{\upsilon }}\right|d\mathsf{\upsilon }=\underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\right|d\mathsf{\upsilon }<\infty .$$

(13)

□

This theorem is fundamental in the field of signal analysis and system theory. Because it states that a function that is in $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right),$ its frequency spectrum is decomposed dyadically, obtaining different combinations of sinusoidal frequencies, which is essential for the analysis of the frequency components of a function.

Definition 2

(${\mathcal{T}}_e$ Transform in Two Parameters). Given $f\left(\upsilon \right)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right),$ the ${\mathcal{T}}_e$ transform in two parameters of $f\left(\upsilon \right)$ is defined as shown in Equation (12).

Theorem 2

(${\mathcal{T}}_e$ Transform $\in L^1\left(\mathbb{R}\right)$). If $f\left(\upsilon \right)\in L^1\left(\mathbb{R}\right),$ then ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\in L^1\left(\mathbb{R}\right)$.

Proof. 

To carry out the demonstration, it is enough to prove that ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)$ converges punctually. To do this, we are going to use Lebesgue’s dominated convergence theorem, shown in Section 2. For this, we calculate the ${\mathcal{T}}_e$ transform of $f\left(\upsilon \right)$ around the angular frequency $\chi$, as shown in Equation (14).

$${\mathcal{T}}_{\mathrm{e}}\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\left\{\omega +\mathsf{\chi }\right\}\right)-{\mathcal{T}}_{\mathrm{e}}\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\left[e^{-i\left(\omega +\mathsf{\chi }\right)\mathsf{\upsilon }}-e^{-i\omega \mathsf{\upsilon }}\right]d\mathsf{\upsilon }.$$

(14)

Rearranging the integrand, we obtain Equation (15).

$$\underset{-\infty }{\overset{\infty }{\int }}{f}_{\mathsf{\chi }}\left(\mathsf{\upsilon }\right)d\mathsf{\upsilon }=\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}\left[e^{-i\mathsf{\chi }\mathsf{\upsilon }}-1\right]d\mathsf{\upsilon },$$

(15)

where $f_{\chi }\left(\upsilon \right)=f\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{-i\omega \upsilon }\left[e^{-i\chi \upsilon }-1\right]$. Subsequently, $\underset{\chi \to 0}{\mathit{lim}}{f}_{\chi }\left(\upsilon \right)$ is calculated, see Equation (16).

$$\underset{\mathsf{\chi }\to 0}{\lim }{f}_{\mathsf{\chi }}\left(\mathsf{\upsilon }\right)=f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}\underset{\mathsf{\chi }\to 0}{\lim }\left(e^{-i\mathsf{\chi }\mathsf{\upsilon }}-1\right)=0.$$

(16)

After calculating $\underset{\chi \to 0}{\mathit{lim}}{f}_{\chi }\left(\upsilon \right)$ and because $\left|f\left(\upsilon \right)\right|$ is integrable using the dominated convergence theorem, we obtain that ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)\in L^1\left(\mathbb{R}\right)$. □

The main importance of having the ${\mathcal{T}}_e$ transform in $L^1\left(\mathbb{R}\right)$ is that it allows analyzing functions in terms of their spectral content, revealing what the composition of the function is like in the dyadic frequency domain to then be able to recover said function.

Theorem 3

(Existence of the Inverse ${\mathcal{T}}_{\mathrm{e}}$ Transform). If $f\left(\upsilon \right)\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$, then Equation (17) is the inverse ${\mathcal{T}}_e$ transform.

$$f\left(\upsilon \right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{i\omega \upsilon }d\omega d\mu .$$

(17)

Proof. 

To perform the proof, we first calculate the Fourier transform of ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)$ for $\mu .$

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }{e}^{-i\chi \mu }d\mu ,$$

(18)

where:

• $\chi$ is the angular frequency associated with $\mu$,
• $\mathcal{F}$ represents the operator of the Fourier transform, and we will use this notation whenever we depict the Fourier transform of the ${\mathcal{T}}_e$ transform.

Applying Fubini’s theorem, we obtain Equation (19).

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\chi \mu }d\mu d\mathsf{\upsilon }.$$

(19)

Since the ${\mathcal{T}}_e$ window is even and holds the scaling property of a dyadic wavelet function, the Fourier transform of ${\beta }_{\mu ,\xi }\left(\upsilon \right)$ associated with $\mu$ is calculated as follows:

$$B\left(\upsilon ,\chi \right)=\frac{1}{\sqrt{2^{\xi }}}\underset{-\infty }{\overset{\infty }{\int }}\beta \left(\mu -\frac{\upsilon }{2^{\xi }}\right)e^{-i\chi \mu }d\mu .$$

We make a change of variable $\mu -\frac{\upsilon }{2^{\xi }}=\tau ,$ obtaining $\mu =\tau +\frac{\upsilon }{2^{\xi }}$ and $d\mu =d\tau$:

$$B\left(\upsilon ,\chi \right)=\frac{1}{\sqrt{2^{\xi }}}{e}^{-i\chi \frac{\upsilon }{2^{\xi }}}\underset{-\infty }{\overset{\infty }{\int }}\beta \left(\tau \right)e^{-i\chi \tau }d\tau ,$$

$$B\left(\upsilon ,\chi \right)=\frac{1}{\sqrt{2^{\xi }}}{e}^{-i\chi \frac{\upsilon }{2^{\xi }}}B\left(\chi \right).$$

(20)

If we use the result obtained in Equation (20) in Equation (19), we obtain Equation (21). Those steps are shown below.

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right)\frac{1}{\sqrt{2^{\xi }}}{e}^{-i\chi \frac{\upsilon }{2^{\xi }}}B\left(\chi \right)e^{-i\omega \upsilon }d\upsilon ,$$

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\frac{1}{\sqrt{2^{\xi }}}\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right)B\left(\chi \right)e^{-i\left[\omega +\frac{\chi }{2^{\xi }}\right]\upsilon }d\upsilon ,$$

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\frac{1}{\sqrt{2^{\xi }}}B\left(\chi \right)\underset{-\infty }{\overset{\infty }{\int }}f\left(\upsilon \right)e^{-i\left[\omega +\frac{\chi }{2^{\xi }}\right]\upsilon }d\upsilon ,$$

$$\mathcal{F}\left\{{\mathcal{T}}_e\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)\right\}=\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\mathrm{B}\left(\chi \right)\mathcal{F}\left[f\left(\mathsf{\upsilon }\right)\right]\left(\omega +\frac{\chi }{2^{\xi }}\right).$$

(21)

Then, we apply Parseval’s theorem to the integral that is between the square brackets, obtaining Equation (23).

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\left[\underset{-\infty }{\overset{\infty }{\int }}{\mathcal{T}}_e\left[f\left(\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)d\mu \right]e^{i\omega \mathsf{\upsilon }}d\omega ,$$

(22)

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\left[\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\mathrm{B}\left(\chi \right)\mathcal{F}\left[f\left(\mathsf{\upsilon }\right)\right]\left(\omega +\frac{\chi }{2^{\xi }}\right){\left[\frac{1}{\sqrt{2^{\mathsf{\xi }}}}{e}^{-i\chi \frac{\upsilon }{2^{\xi }}}\mathrm{B}\left(\chi \right)\right]}^{\ast }d\chi \right]e^{i\omega \mathsf{\upsilon }}d\omega ,$$

(23)

where * is the complex conjugate operator.

Rewriting Equation (23), we obtain Equation (24).

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\left[\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\mathrm{F}\left(\omega +\frac{\chi }{2^{\xi }}\right)\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\mathrm{B}\left(\chi \right)\frac{1}{\sqrt{2^{\mathsf{\xi }}}}{\mathrm{B}}^{\ast }\left(\chi \right)e^{i\chi \frac{\upsilon }{2^{\xi }}}d\chi \right]e^{i\omega \mathsf{\upsilon }}d\omega .$$

(24)

Since $\frac{1}{\sqrt{2^{\xi }}}\in \mathbb{R}$, this means that $\frac{1}{\sqrt{2^{\xi }}}B\left(\chi \right)\frac{1}{\sqrt{2^{\xi }}}{B}^{\mathrm{*}}\left(\chi \right)$ can be rewritten as ${\left|B^{’}\left(\chi \right)\right|}^2$.

Rewriting Equation (24) and applying Fubini’s theorem (see Section 2), we obtain Equation (25).

$$f\left(\mathsf{\upsilon }\right)={\left(\frac{1}{2\pi }\right)}^2\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\mathrm{F}\left(\omega +\frac{\chi }{2^{\xi }}\right){\left|{\mathrm{B}}^{\prime }\left(\chi \right)\right|}^2{e}^{i\left[\omega +\frac{\chi }{2^{\xi }}\right]\mathsf{\upsilon }}d\omega d\chi$$

(25)

Then, we organize the integral of Equation (25) to express it as observed in Equation (26).

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\left[\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\mathrm{F}\left(\omega +\frac{\chi }{2^{\xi }}\right)e^{i\left({\mathsf{\omega }+2}^{-\xi }\chi \right)\mathsf{\upsilon }}d\omega \right]{\left|{\mathrm{B}}^{\prime }\left(\chi \right)\right|}^{2}d\chi$$

(26)

To solve Equation (26), we first make a change of variable to the integral inside the square brackets ${\omega }^{’}=\omega +2^{-\xi }\chi$ and $d{\omega }^{´}=d\omega +d\left(2^{-\xi }\chi \right)$ obtaining $d{\omega }^{´}=d\omega$. Equation (27) shows this integral with the change of variables. Observe that we are in the presence of the inverse Fourier transform of the function $F\left({\omega }^{´}\right)$, obtaining $f\left(\upsilon \right)$.

$$f\left(\upsilon \right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F\left({\omega }^{´}\right)e^{i{\omega }^{´}\upsilon }d{\omega }^{´}.$$

(27)

If we substitute the result of Equation (27) in Equation (26), we obtain Equation (28).

$$f\left(\mathsf{\upsilon }\right)=\frac{1}{2\mathsf{\pi }}\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right){\left|{\mathrm{B}}^{\prime }\left(\chi \right)\right|}^{2}d\chi .$$

(28)

Then, we rewrite Equation (28) obtaining Equation (29),

$$f\left(\mathsf{\upsilon }\right)=\frac{f\left(\mathsf{\upsilon }\right)}{2\mathsf{\pi }}\underset{-\infty }{\overset{\infty }{\int }}{\left|{\mathrm{B}}^{\prime }\left(\chi \right)\right|}^{2}d\chi .$$

(29)

Through reference [9] and applying Plancherel’s theorem, we obtain Equation (30).

$$\underset{-\infty }{\overset{\infty }{\int }}{\left|{\beta }_{\mu ,\xi }\left(\upsilon \right)\right|}^{2}d\mu =\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{\left|B^{’}\left(\chi \right)\right|}^{2}d\chi =1.$$

(30)

Then, the identity $f\left(\upsilon \right)=f\left(\upsilon \right)$ holds, which indicates that the inverse ${\mathcal{T}}_e$ transform exists. □

Having a tool that allows us to recover a function from the dyadic frequency domain is highly valuable in system analysis. This is because it simplifies function analysis through dyadic frequency representation, particularly when dealing with complex or noisy functions. Additionally, it enables noise reduction by isolating and eliminating unwanted or spurious frequency components.

Theorem 4

(Uniqueness). Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be two functions for which there are their ${\mathcal{T}}_e$ transforms. Then, ${\mathcal{T}}_e\left[f\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)={\mathcal{T}}_e\left[g\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)$if and only if $f\left(\upsilon \right)=g\left(\upsilon \right)$ almost everywhere.

Proof. 

Necessity: To carry out the demonstration, we first calculate the magnitude of the difference of the ${\mathcal{T}}_e$ transform for $f\left(\upsilon \right)$ and $g\left(\upsilon \right)$. Finally, we use Theorem 3 (Existence of the inverse ${\mathcal{T}}_e$ transform) to use the inverse ${\mathcal{T}}_e$ transform.

$$\left|{\mathcal{T}}_e\left[f\left(\mathsf{\upsilon }\right)\right]-{\mathcal{T}}_e\left[g\left(\mathsf{\upsilon }\right)\right]\right|\left(\mu ,2^{\xi }\omega \right)=\left|\underset{-\infty }{\overset{\infty }{\int }}f\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }-\underset{-\infty }{\overset{\infty }{\int }}g\left(\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }\right|,$$

$$\left|\underset{-\infty }{\overset{\infty }{\int }}\left[f\left(\mathsf{\upsilon }\right)-g\left(\mathsf{\upsilon }\right)\right]{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }\right|\le \underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right)-g\left(\mathsf{\upsilon }\right)\right|\left|{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\right|\left|e^{-i\omega \mathsf{\upsilon }}\right|d\mathsf{\upsilon }.$$

Since $\left|e^{-i\omega \mathsf{\upsilon }}\right|=1,$ then

$$\left|\underset{-\infty }{\overset{\infty }{\int }}\left[f\left(\mathsf{\upsilon }\right)-g\left(\mathsf{\upsilon }\right)\right]{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }\right|\le \underset{-\infty }{\overset{\infty }{\int }}\left|f\left(\mathsf{\upsilon }\right)-g\left(\mathsf{\upsilon }\right)\right|\left|{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)\right|d\mathsf{\upsilon }=0.$$

Sufficiency: To prove the other sense of equivalence, we will use the shown linearity property [2] for the ${\mathcal{T}}_e$ transform and the inverse ${\mathcal{T}}_e$ transform theorem shown in Theorem 3:

$$f\left(\upsilon \right)-g\left(\upsilon \right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{\mathcal{T}}_e\left[f\left(\upsilon \right)-g\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{i\omega \upsilon }d\omega d\mu =0.$$

Since ${\mathcal{T}}_e\left[f\left(\upsilon \right)-g\left(\upsilon \right)\right]\left(\mu ,2^{\xi }\omega \right)={\int }_{-\infty }^{\infty }f\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{-i\omega \upsilon }d\upsilon -{\int }_{-\infty }^{\infty }g\left(\upsilon \right){\beta }_{\mu ,\xi }\left(\upsilon \right)e^{-i\omega \upsilon }d\upsilon =0$. □

The uniqueness theorem in the ${\mathcal{T}}_e$ transform is a fundamental property that states that a function in the time domain is completely determined by its ${\mathcal{T}}_e$ transform in the dyadic frequency domain. From an applied mathematics point of view, such a theorem guarantees that, given the dyadic frequency spectrum of a function, it is possible to recover the original function in the time domain in a unique way. This is crucial in applications where functions need to be reconstructed from their spectrum, such as in communication, signal processing, and data retrieval.

4. Examples of Transformed Pairs

• ${\mathcal{T}}_{\mathit{e}}$ Transform of the constant $\mathit{f}\left({\mathit{\omega }}_{\mathit{c}}\mathsf{\upsilon }\right)$ where ${\mathit{\omega }}_{\mathit{c}}=0$:

$${\mathcal{T}}_e\left[f\left(0\right)\right]\left(\mu ,2^{\xi }\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(0\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon },$$

$${\mathcal{T}}_e\left[f\left(0\right)\right]\left(\mu ,2^{\xi }\omega \right)=f\left(0\right)\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }.$$

We make a change of variable $\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }=\tau ,$ obtaining $\mathsf{\upsilon }=2^{\xi }\left(\tau +\mu \right)$ and $d\upsilon =2^{\mathsf{\xi }}d\tau$,

$${\mathcal{T}}_e\left[f\left(0\right)\right]\left(\mu ,2^{\xi }\omega \right)=2^{\mathsf{\xi }}f\left(0\right)\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\omega 2^{\xi }\left(\tau +\mu \right)}d\tau ,$$

$${\mathcal{T}}_e\left[f\left(0\right)\right]\left(\mu ,2^{\xi }\omega \right)=f\left(0\right)\sqrt{2^{\xi }}{e}^{-i\omega 2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\omega 2^{\xi }\tau }d\tau ,$$

$${\mathcal{T}}_e\left[f\left(0\right)\right]\left(\mu ,2^{\xi }\omega \right)=f\left(0\right)\sqrt{2^{\xi }}{e}^{-i\omega 2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\mathsf{\omega }\right).$$

where $f\left(0\right)$ is a constant and $\mathrm{B}\left(2^{\xi }\mathsf{\omega }\right)$ is the dyadic frequency response of $\mathsf{\beta }\left(\tau \right)$ for ${\omega }_c=0$ and it is the angular center frequency.

Figure 3 shows the frequency spectrum obtained for the ${\mathcal{T}}_e$ transform of the function $f\left(0\right)$. Note in this figure that there is a frequency spectrum for each value of $\xi$, obtaining a ${\mathcal{T}}_e$ window for each scale ($2^{\xi }$ where $\xi \in \left[13\right]$). The authors call this result a dyadic frequency spectrum because $\mathsf{\omega }$ is dilated by a factor of $2^{\xi }$.

• ${\mathcal{T}}_{\mathit{e}}$  Transform of the complex exponential:

$${\mathcal{T}}_e\left[e^{i{\omega }_c\mathsf{\upsilon }}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{1}{\sqrt{2^{\mathsf{\xi }}}}\underset{-\infty }{\overset{\infty }{\int }}{e}^{i{\omega }_c\mathsf{\upsilon }}{\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon },$$

$${\mathcal{T}}_e\left[e^{i{\omega }_c\mathsf{\upsilon }}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{1}{\sqrt{2^{\xi }}}\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\left(\omega -{\omega }_c\right)\mathsf{\upsilon }}d\mathsf{\upsilon }.$$

We make a change of variable $\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }=\tau ,$ obtaining $\mathsf{\upsilon }=2^{\xi }\left(\tau +\mu \right)$ and $d\upsilon =2^{\mathsf{\xi }}d\tau$,

$${\mathcal{T}}_e\left[e^{-i{\omega }_c\mathsf{\upsilon }}\right]\left(\mu ,2^{\xi }\omega \right)=\sqrt{2^{\xi }}\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\left(\tau +\mu \right)}d\tau ,$$

$${\mathcal{T}}_e\left[e^{i{\omega }_c\mathsf{\upsilon }}\right]\left(\mu ,2^{\xi }\omega \right)=\sqrt{2^{\xi }}{e}^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\beta \left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\tau }d\tau ,$$

$${\mathcal{T}}_e\left[e^{i{\omega }_c\mathsf{\upsilon }}\right]\left(\mu ,2^{\xi }\omega \right)={\sqrt{2^{\xi }}e}^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\left[\omega -{\omega }_c\right]\right).$$

Figure 4 shows the dyadic frequency spectrum obtained after calculating the ${\mathcal{T}}_e$ transform to the function $e^{i{\omega }_c\mathsf{\upsilon }}$. Note that the frequency spectrum does not distort the frequency component, showing how it can be observed at different scales. For this case, we can see how if we multiply the ${\mathcal{T}}_e$ window by $e^{i{\omega }_c\mathsf{\upsilon }}$, we obtain a translation of the window for ${\omega }_c$. Furthermore, since the window has a limited spectral bandwidth, we can observe how this bandwidth is modified by the scale, showing the existence of different bandwidths that indicate sensitive behavior for different frequency components, known as multi-sensitivity.

• ${\mathcal{T}}_{\mathit{e}}$ Transform of the sine function:

$${\mathcal{T}}_e\left[\sin \left({\omega }_c\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}\sin \left({\omega }_c\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon },$$

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}-e^{-i{\omega }_c\mathsf{\upsilon }}}{2i}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{1}{\sqrt{2^{\xi }}2i}\left[\underset{-\infty }{\overset{\infty }{\int }}{e}^{i{\omega }_c\mathsf{\upsilon }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }-\underset{-\infty }{\overset{\infty }{\int }}{e}^{-i{\omega }_c\mathsf{\upsilon }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }\right],$$

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}-e^{-i{\omega }_c\mathsf{\upsilon }}}{2i}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{1}{\sqrt{2^{\xi }}2i}\left[\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\left(\omega -{\omega }_c\right)\mathsf{\upsilon }}d\mathsf{\upsilon }-\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\left(\omega +{\omega }_c\right)\mathsf{\upsilon }}d\mathsf{\upsilon }\right],$$

We make a change of variable $\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }=\tau$, obtaining $\mathsf{\upsilon }=2^{\xi }\left(\tau +\mu \right)$ and $d\upsilon =2^{\mathsf{\xi }}d\tau$,

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}-e^{-i{\omega }_c\mathsf{\upsilon }}}{2i}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{2^{\mathsf{\xi }}}{\sqrt{2^{\xi }}2i}\left[\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\left(\tau +\mu \right)}d\tau -\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\left(\tau +\mu \right)}d\tau \right],$$

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}-e^{-i{\omega }_c\mathsf{\upsilon }}}{2i}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{\sqrt{2^{\xi }}}{2i}\left[e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\beta \left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\tau }d\tau -e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\beta \left(\tau \right)e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\tau }d\tau \right],$$

$${\mathcal{T}}_e\left[\sin \left({\omega }_c\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=i\sqrt{2^{\xi -2}}\left[-e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\left[\omega -{\omega }_c\right]\right)+e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\left[\omega +{\omega }_c\right]\right)\right].$$

Figure 5 shows the ${\mathcal{T}}_e$ transform of the sine function. Notice how the result obtained is similar to that of the Fourier transform of a sine function [26,27,28]. This is because the ${\mathcal{T}}_e$ transform is derived from the Fourier transform. For this reason, we observe how we have a bilateral and odd spectrum. An evident advantage of the ${\mathcal{T}}_e$ transform over the Fourier transform and even the short-time Fourier transform [29,30] is the possibility of dyadic scaling of the frequency spectrum of the sine function. This allows a dyadic representation of said function to be obtained.

This example reveals how the dyadic frequency spectrum preserves the bilateral frequency spectrum, now called the bilateral dyadic frequency spectrum. The fact of having a window function that contracts in the frequency domain will allow discriminating frequency components over ${\mathsf{\omega }}_{\mathrm{c}}$.

• ${\mathcal{T}}_{\mathit{e}}$  Transform of the cosine function:

Notice how in this example a very similar result to the one obtained for the ${\mathcal{T}}_e$ transform of the sine function is obtained.

$${\mathcal{T}}_e\left[\cos \left({\omega }_c\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}\cos \left({\omega }_c\mathsf{\upsilon }\right){\beta }_{\mu ,\xi }\left(\mathsf{\upsilon }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon },$$

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}+e^{-i{\omega }_c\mathsf{\upsilon }}}{2}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{1}{\sqrt{2^{\xi }}2}\left[\underset{-\infty }{\overset{\infty }{\int }}{e}^{i{\omega }_c\mathsf{\upsilon }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }+\underset{-\infty }{\overset{\infty }{\int }}{e}^{-i{\omega }_c\mathsf{\upsilon }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\omega \mathsf{\upsilon }}d\mathsf{\upsilon }\right],$$

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}+e^{-i{\omega }_c\mathsf{\upsilon }}}{2}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{\sqrt{2^{\xi }}}{2^{\xi }2}\left[\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\left(\omega -{\omega }_c\right)\mathsf{\upsilon }}d\mathsf{\upsilon }+\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }\right)e^{-i\left(\omega +{\omega }_c\right)\mathsf{\upsilon }}d\mathsf{\upsilon }\right].$$

We make a change of variables $\frac{\mathsf{\upsilon }}{2^{\mathsf{\xi }}}-\mathsf{\mu }=\tau$, obtaining $\mathsf{\upsilon }=2^{\xi }\left(\tau +\mu \right)$ and $d\upsilon =2^{\mathsf{\xi }}d\tau$,

$${\mathcal{T}}_e\left[\frac{e^{i{\omega }_c\mathsf{\upsilon }}+e^{-i{\omega }_c\mathsf{\upsilon }}}{2}\right]\left(\mu ,2^{\xi }\omega \right)=\frac{\sqrt{2^{\xi }}}{2^{\xi }2}{2}^{\mathsf{\xi }}\left[\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\left(\tau +\mu \right)}d\tau +\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\left(\tau +\mu \right)}d\tau \right],$$

$${\mathcal{T}}_e\left[\cos \left({\omega }_c\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=\sqrt{2^{\xi -2}}\left[e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega -{\omega }_c\right)2^{\mathsf{\xi }}\tau }d\tau +e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\mu }\underset{-\infty }{\overset{\infty }{\int }}\mathsf{\beta }\left(\tau \right)e^{-i\left(\omega +{\omega }_c\right)2^{\mathsf{\xi }}\tau }d\tau \right],$$

$${\mathcal{T}}_e\left[\cos \left({\omega }_c\mathsf{\upsilon }\right)\right]\left(\mu ,2^{\xi }\omega \right)=\sqrt{2^{\xi -2}}\left[e^{-i\left(\omega -{\omega }_c\right)2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\left[\omega -{\omega }_c\right]\right)+e^{-i\left(\omega +{\omega }_c\right)2^{\xi }\mu }\mathrm{B}\left(2^{\xi }\left[\omega +{\omega }_c\right]\right)\right].$$

Figure 6 shows the dyadic frequency spectrum for a cosine function. It is worth noting how this frequency spectrum retains the parity characteristic observed in the frequency spectrum obtained through the Fourier transform [28,31], adding the advantage of being able to analyze the frequency component of the function in different scales.

5. Conclusions

This paper presents six contributions to knowledge in the field of mathematics applied to engineering. The main objective of these contributions is to show the mathematical foundations for creating an environment on the ${\mathcal{T}}_e$ domain.

The combination of the two windows forming the ${\mathcal{T}}_e$ window provides maximum noise attenuation. Being able to have a window like the one mentioned above allows the ${\mathcal{T}}_e$ transform to have an inverse. This means that an absolutely integrable function can be recovered to which the ${\mathcal{T}}_e$ transform has been applied, making its use more attractive like an integral transform.

Recovering a function from its integral representation in the dyadic frequency spectrum is of great importance in various fields of science, engineering, and technology, such as audio and music processing, engineering, data processing, and time series analysis, among others.

On the other hand, having a new definition of the ${\mathcal{T}}_e$ transform in two parameters maximizes the precision of mathematical analysis. Furthermore, it provides a new bi-dimensional approach for analyzing functions, referred to as scale frequency. This approach allows us to understand how scale varies as a function of frequency.

The uniqueness theorem emphasizes the significant importance of making it clear that the recovery of a function from its dyadic frequency spectrum is only possible through the ${\mathcal{T}}_e$ transform.

From the examples, we observe that having a multi-sensitive spectrum (dyadic frequency spectrum) enables the isolation of frequency components, allowing for their identification. This strengthens applications aimed at fault diagnosis, which is crucial in fields such as mechanical engineering and seismic analysis.

Having the ${\mathcal{T}}_e$ transform in ${\mathrm{L}}^1\left(\mathbb{R}\right)$ allows us to validate recovery theorems and all the theorems that rely on the ${\mathcal{T}}_e$ transform inversion theorem. In applications where it is desirable to modify spectral characteristics of functions, it is of great importance to subsequently return the function to the time or spatial domain.

In future studies, we will investigate the energy preservation of a signal after applying the ${\mathcal{T}}_e$ transform. We will delve into the reconstruction of functions from the ${\mathcal{T}}_e$ transform and assess its error. We will also conduct a computational complexity analysis to determine its effectiveness in noise attenuation within the fields of engineering and technology. Furthermore, we will employ the ${\mathcal{T}}_e$ transform in fault detection applications.