The convolution integral equations are very important in optics and signal processing domain. In this paper, fractional mixed-weighted convolution is defined based on the fractional cosine transform; the corresponding convolution theorem is achieved. The 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.

中文翻译：

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分数阶混合加权卷积及其在卷积积分方程中的应用

卷积积分方程在光学和信号处理领域非常重要。本文基于分数余弦变换定义了分数混合加权卷积；得到相应的卷积定理。还探讨了分数混合加权卷积和杨氏类型定理的性质。基于分数阶混合加权卷积和分数余弦变换，考虑了两种卷积积分方程，得到了分数阶卷积积分方程的显式解，并分析了解的计算复杂度。