Research On Prony Approximations Of Several Special Functions

As a spectrum estimation method,Prony's method is widely used in various inverse problems,such as power system response signal analysis,optimization of sparse antenna arrays,and ultrasonic signal analysis.In this paper,we explore the application of Prony's method in the field of numerical approximation.We use Prony exponential approxima-tion and Prony-like triangular series approximation to study exponential integral,cosine integral,sine integral,and sinc function.In addition,we optimize the Prony-like method for Bessel functions of the first kind of integer order.More specifically,1.We study Prony's method based on generalized eigenvalues and two kinds of Prony-like methods in trigonometric function form.2.By analyzing the properties of exponential integration,cosine integration,sine integration,and sinc function,we apply Prony's method to approximate these special functions.The experimental results of Prony's method are compared with traditional numerical approximation methods such as:power series expansion,asymptotic series expansion,continuous fraction expansion,etc.The numerical experimental results in Maple show that the Prony's method has better approximation accuracy and can maintain a stable approximation effect throughout the interval.3.We optimize the Prony-like method for Bessel functions of the first kind of integer order,while improving the calculation speed and ensuring the approximation accuracy.On the one hand,we use Chebyshev nodes to replace the parameter?_i,which avoids the calculation of generalized eigenvalues,reduces the computational complexity,and improves the efficiency of approximation.On the other hand,we use the interval sampling method to optimize the process of solving the parameter?_i for overdetermined linear equations,while improving the approximation efficiency and ensuring the accuracy of evaluation.