Two basic dual trigonometric series relations involving a countable infinite number of unknowns are considered for the determination of the unknowns. The numerical values of the unknowns are determined with the help of the methods of algebraic least-squares approximation and singular value decomposition. The dual trigonometric series and the corresponding functions are compared with the existing results. The errors are also computed to show the efficiency of these methods. The study indicates that the method of algebraic least squares is more straightforward, simpler and computationally more efficient as compared to the available methods.

中文翻译：

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求解对偶三角级数关系的改进方法

为了确定未知数，考虑了涉及可数无限数量的未知数的两个基本的双三角级数关系。未知数的数值借助于代数最小二乘逼近和奇异值分解的方法来确定。对偶三角级数和相应的函数与现有结果进行了比较。还计算了误差以显示这些方法的效率。研究表明，与可用方法相比，代数最小二乘方法更直接，更简单且计算效率更高。