Fractional Mixed Weighted Convolution and Its Application in Convolution Integral Equations

. Te convolution integral equations are very important in optics and signal processing domain. In this paper, fractional mixed-weighted convolution is defned based on the fractional cosine transform; the corresponding convolution theorem is achieved. Te properties of fractional mixed-weighted convolution and Young’s type theorem are also explored. Based on the fractional mixed-weighted convolution and fractional cosine transform, two kinds of convolution integral equations are considered, the explicit solutions of fractional convolution integral equations are obtained, and the computational complexity of solutions are also analyzed.


Introduction
Te convolution is a powerful mathematical tool that plays a crucial role in various felds such as applied mathematics, harmonic analysis, integral equation solving, signal processing, image processing, and neural networks [1][2][3][4][5][6].It enables signal fltering, feature extraction, and system response analysis functions, making it highly signifcant for advanced signal processing and pattern recognition realization.
Te convolution integral equations arise in many branches of natural science and have important applications in various felds such as engineering mechanics, dynamic theory, mathematical theory of the spatial-temporal spread of pandemic, especially when solving the problems of optical systems and the digital signal processing domain [7][8][9][10][11][12][13]. In recent years, the convolution integral equations have been studied extensively by many researchers [14][15][16][17][18][19][20].Tuan [14] studied solvability in close form and estimated the boundedness solutions of some classes for integral diferential equations of the Barbashin type and the Frcdhohn type integral equation.Askhabov [15] studied various classes of nonlinear convolution-type integral equations appearing in the theory of feedback systems.Sun et al. [16] studied the existence and noethericity of solutions for two classes of singular convolution integral equations with Cauchy kernels in the nonnormal type case.And the solutions for some singular convolution integral equations are discussed in [17][18][19][20].
However, all the studies mentioned above are based on Fourier analysis theory, which seriously limits its application scope because of its nonlocality.Tese limitations force people to fnd some improvement methods.In recent years, many scholars devote themselves to extending Fourier analysis to fractional domain and study fractional convolution integral equations.K. Razminia and A. Razminia [21] studied fractional difusion equation (FDE) using the convolution integral.Li et al. [22] analyzed the solvability of the convolution equations by convolution operator for a twodimensional fractional Fourier transform in polar coordinates.Feng and Wang [23] discussed explicit solutions of the convolution-type integral equations using the generalized fractional convolution.In [24,25], the author studied the convolution-type integral equations based on the fractional Laplace convolutions.As far as we are concerned, on the one hand, compared with the wide applications of convolution equation in the Fourier domain, the fractional convolution equation is less studied.On the other hand, the mixed-weighted convolution equations in fractional domain have not been studied yet.Hence, it is therefore interesting and worthwhile to investigate fractional convolution integral equations in depth, and how to obtain solutions to these convolution equations is one of the meaningful issues of equation theory.
In this paper, we investigate two types of fractional mixed-weighted convolution integral equations.Te contribution of this study is threefold: (1) we propose two kinds of fractional mixed-weighted convolution that enable the processing and analysis of input data by selecting appropriate weighting coefcients, thereby achieving the desired processing efect.(2) Te proposed fractional mixedweighted convolution for the fractional cosine transform can be expressed by classical convolution.(3) We apply these newly developed convolution structures to discuss solutions for the convolution integral equations, which can be efciently computed using FFT and which exhibit lower computational complexity compared to methods employed in the FRFT domain.
Te rest of this paper is organized as follows: Section 2 presents preliminaries.In Section 3, the fractional mixedweighted convolution for the fractional cosine transform is proposed, and the corresponding convolution theorem is derived.Te important relation between the mixedweighted fractional convolution and the classical convolution is established, and properties and Young's type theorem are further investigated.In Section 4, two kinds of fractional mixed-weighted convolution equations are discussed, explicit solutions for these convolution equations are given, and the computation complexity of solutions are analyzed.Conclusions are summarized in Section 5.

Preliminaries
In this section, we introduce some defnitions and important properties of the fractional Fourier transform, fractional cosine transform, and corresponding convolution operation.
where (F c f)(u) denotes the FCT.From ( 5) and ( 6), the FRCT can be expressed by FCT as follows: and 7), we can realize the calculation of FRCT (see Figure 1).For N point of samples, FRCT has the same computational complexity as FCT, that is, O(1/2N log N 2 ), which is very important in practical applications.
Te classical convolution operation [30] is given by which satisfes the following convolution theorem: where * denotes the classical convolution operation.Te fractional cosine convolution, denoted by (h * F α c f)(t) was recently defned in [28].
and the corresponding convolution theorem for FRCT is satisfed

Fractional Mixed-Weighted Convolution and Convolution Theorem for FRCT
3.1.Fractional Mixed-Weighted Convolution for FRCT.In this subsection, we give fractional mixed-weighted convolution for the fractional cosine transform, the relationship between proposed convolution and classical convolution is given.

Defnition 1. For any two functions
, fractional mixed-weighted convolution operation of h(t) and f(t) for fractional cosine transform is defned as follows: e jt 2 /2cotφ f(t), and Based on the Defnition 1, the fractional mixed-weighted convolution operation for the FRCT can be implemented in Figure 2.
) and f(t) ∈ L 1 (R + ), then the fractional mixed-weighted convolution operations Based on Defnition 1, we will give the fractional mixedweighted convolution theorem associated with the fractional cosine transform.

Fractional Mixed-Weighted Convolution Teorem for FRCT.
In this section, the fractional mixed-weighted convolution theorem for the fractional cosine transform is derived.
Proof.We frst prove the existence of the convolution op- Figure 1: Te calculation process of FRCT.

Journal of Mathematics
Since Te same estimation is obtained for the other three integrals in a similar manner According to ( 12), ( 13), (16), and ( 17), we have . Next, we prove the convolution Teorem 4. We have therefore, it follows from equations ( 12), (20), and ( 21), we can obtain

4
Journal of Mathematics Tis completes the proof.Te fractional mixed-weighted convolution is very diffcult to implement in the time domain due to the integral operation, as it is evident from Defnition 1 and Figure 2.However, thanks to Teorem 4, it can be realized in the FRCT domain (refer to Figure 3).For N points of samples, the computational complexity of the fractional mixedweighted convolution is given by O(2N log N 2 ).
Remark 6. Te fractional mixed-weighted convolution theorem in equation ( 15) preserves the convolution property for the classical Fourier transform, meaning that the fractional mixed-weighted convolution of two functions is equivalent to multiplying their FT and FRCT.Tis can be particularly useful in solving convolution integral equations and designing flters.

Properties of Fractional Mixed-Weighted Convolution.
In this subsection, properties of fractional mixed-weighted convolution are given and the corresponding Young's type theorem is also explored as follows.
Theorem .Te fractional mixed-weighted convolution for FRCT is not commutative or associative, but is distributive and linear, which satisfes the following equations: (1) Proof.Te distributivity and linearity can be proven by Defnition 1 and Teorem 4, therefore, they are omitted here.

Application of Mixed-Weighted Convolution in the Integral Equation
Te convolution integral equation is of great importance in various applications, particularly in solving engineering problems such as optical systems and digital signal processing.Tese problems can be transformed into the forms of ( 35) and ( 43).How to solve the solutions of these equations is one of the meaningful issues of equation theory.
Next, we will use the convolution theorem derived in this paper to study two types of convolution integral equations.

Te First Kind of the Convolution Integral Equation.
In this subsection, we shall focus on the following convolution integral equation: where are given, and h is unknown function.After simplifcation, (35) can be rewritten in the following form: where (φ * α c h)(t) denotes the fractional mixed-weighted convolution operation in (12), and (h * [28].By applying fractional cosine transform to both sides of (36) and utilizing (15) and Teorem 7 (refer to [28]), we can obtain where Case 9. When λ 1 ≠ 0 and λ 2 , λ 3 are not all zero, from [31], there exists a constant C > 0, such that λ 1 + W(u) ≠ 0, for all u > C. Hence, 1/(λ 1 + W(u)) is bounded and continuous, and we have Applying inverse transform of FRCT to equation (34), we can obtain the general solution of equation (32) as follows: Case 10.When λ 1 � 0 and λ 2 , λ 3 are not all zero, for all u > 0, such that W(u) ≠ 0, the general solution of equation ( 32) is obtained in a similar manner as described as follows: From the above analysis, we give the main results about the solution of (35).32) has the general solution as follows: (1) When λ 1 ≠ 0 and λ 2 , λ 3 are not all zero, for all u > C > 0. Ten, the solution of ( 35) is given by (2) When λ 1 � 0 and λ 2 , λ 3 are not all zero, for all u > 0, then the solution of ( 35) is given by

Te Second Kind of System of the Convolution Integral
be unknown functions, we consider system of convolution integral (43) as follows: Journal of Mathematics where and  f(t) � f(t)e jt 2 /2cotφ ,  ψ(t) � ψ(t)e jt 2 /2cotφ .D φ and A φ correspond to (2) and ( 12), respectively. where Ten, (43) has the unique solution in L 1 (R + ).
Proof.Te system of convolution integral equation (37) can be rewritten as follows: by applying the fractional cosine transform, (15) and Teorem 7 [28] to both sides of (48), we can obtain According to Wiener-Levi's Teorem [30] and (45), we can derive applying inverse transform of the FRCT to (50), we have 8 Journal of Mathematics Similarly, we get Te proof is completed.

The Complexity Analysis of Solutions to Convolution Integral Equations
Te convolution theorem plays an important role in solving convolution integral equations by allowing for the pointwise multiplication of the transformed known function and kernel function, thereby reducing computational complexity.Now, we provide the computational complexity analysis of the solution to the frst kind of convolution integral (35).
As shown in Figure 4, the solution to (35) can be realized as follows.
We can see that the major computation for the frst kind of convolution integral (35) is mainly focused on calculating G 1 (u) and G 2 (u) due to the mixed-weighted function, where G 1 (u) � 1/(λ 1 + W(u)) and G 2 (u) � 1/W(u).Tis leads to an increase in calculation.However, by using the classical FFT and considering the relationship between FRCT and FT (refer to (6) and Figure 1), we can calculate the complexity of solution of the frst kind of convolution integral (35) is O(5/2Nlog N  2 ) for all λ i ∈ C. Next, let us analyze the computation complexity of the solution achieved in convolution integral (43) in detail.Based on (46) and (47), the solutions h(t) and f(t) of (43) can be implemented in Figures 5 and 6, respectively.
From (46), the solution h(t) can be expressed as the convolution sum, which is difcult to implement in time domain.To simplify calculations, we transform the convolution sum into frequency domain using fractional cosine transform.For a discrete signal of size N, discrete Fourier cosine transform (DFCT) requires a (1/2Nlog N 2 ) real number multiplications.According to (16), Figure 3, and Teorem 7 (see [28]), we can calculate the complexity of (φ * α

Conclusions
Tis paper deals with two kinds of convolution integral equations based on the derived fractional convolution theorem.First, fractional mixed-weighted convolution for the fractional cosine transform is proposed.Second, the corresponding convolution theorem is derived, and properties and Young's type theorem for fractional mixedweighted convolution are studied.Finally, based on the proposed convolution theorem, we discussed two kinds of convolution integral equations and analyzed the computational complexity of the solution of the equation.

Figure 2 :
Figure 2: Implementation of the fractional mixed-weighted convolution in time domain.

Figure 3 :
Figure 3: Implementation of the fractional mixed-weighted convolution in FRCT domain.