一种快速逼近Fourier级数和的实用算子

采用逐次平均方法，构造一种快速逼近Ｆｏｕｒｉｅｒ级数和实用算子（逐次平均算子），用这种算子来取代Ｆｏｕｒｉｅｒ级数无穷项求和，具有误差小，收敛速度快的功能。志Ｆｅｊｅｒ算子比，它的逼近性要好得多，与Ｋｏｒｂｉｎ算子比，它更易于实用，理论证明其一般性，计算著名的Ｓａｆｆｍａｎ－Ｔａｙｌｏｒ粘性指进问题时应用逐次平均算子，既不产生大的截断误差，又能避免高频振荡而使计算过程稳定，得到很好的结果。

A PRACTICAL OPERATOR TO FAST APPROACH FOURIER SERIES SUM

In this paper, we present a practical operator (Successive Averaging ApproximatingOperator) to fast approach Fourier series sum. Summing a finite number of items in an infiniteFourier series expanded from a function is a normal but useful computational method. Anyway,when we sum a Fourier series, there always exists a conflict between accuracy and instability: inorder to reduce the residual error the number of items N in the Fourier series should be as largeas possible. However, large N will introduce high frequency trigonometric functions which alwaysmake numerical computation unstable even collapsing. Traditionally Fejer introduced an operatorbut still with low convergence rate. Another approach is Korbkin operator which has a muchbetter approximation but is difficult to construct. By using Successive Averaging ApproximatingOperator to calculate the famous Saffman-Taylor finger, we reduce the residual errors to a verylow level, making the computation possible and enhancing the calculating speed greatly, which ismuch better than using Fejer Operator, and get very good solution. The generality of SuccessiveAveraging Operator is proved theoretically.