Weighted Convolution In Fractional Domain And Its Applications

As a powerful tool for signal processing,fractional transform plays an important role in applied mathematics,signal and image processing.Convolution is a kind of integral operation,and the convolution of many different integral transform still plays an important role in mathematics and information science.Because different convolution operations have different structures,different complex application scenarios can be modeled in practical applications.Therefore,further study on the convolution and its properties of fractional transform is of great theoretical significance and practical application value for revealing the inherent law of non-stationary signal analysis and processing.In this dissertation,the weighted convolution and its properties of fractional transform are studied on the basis of the existing research results.The main research contents are as follows:1.Fourier weighted convolution and its properties are studied.In this paper,Fourier cosine weighted convolution and Fourier sine weighted convolution are given,the properties of Fourier sine and cosine weighted convolution and Young’s class inequality are studied,and the corresponding weighted convolution theorem is derived.2.The fractional sine and cosine Laplace weighted convolution and its properties are studied.In this dissertation,we extend Fourier sine and cosine transform to fractional domain,and combine it with Laplace transform,define several kinds of fractional Fourier sine and cosine Laplace weighted convolution operations,the relation between the obtained weighted convolution and the existing convolution is given,furthermore,the properties of fractional Fourier sine and cosine Laplace weighted convolution operation and Young’s class inequality are studied.3.The application of fractional Fourier sine and cosine weighted convolution to convolution integral equations is studied.Based on the weighted convolution operation in fractional domain and the corresponding weighted convolution theorem,the solutions of several kinds of convolution integral equations are discussed,and the corresponding explicit solutions are given.