We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such a matrix approximation. When used for computing the Fredholm convolution of two given functions, such approximations produce the convolution more rapidly than the state-of-the-art methods. The proposed approximation also leads to a spectral method for solving the Fredholm convolution integral equations and enables the computation of eigenvalues and pseudospectra of Fredholm convolution operators, which is otherwise intractable with existing techniques.

翻译：我们提出了一种构建Fredholm型卷积算子谱逼近的方法。所提出的算法数值稳定，并利用了此类矩阵逼近元素满足的递推关系。当用于计算两个给定函数的Fredholm卷积时，该逼近方法比现有最优方法更快。所提出的逼近方法还导出了求解Fredholm卷积积分方程的谱方法，并能够计算Fredholm卷积算子的特征值与伪谱，而这是现有技术难以实现的。