This paper studies the cosine as basis function for the approximation of univariate and continuous functions without memory. This work studies a supervised learning to obtain the approximation coefficients, instead of using the Discrete Cosine Transform (DCT). Due to the finite dynamics and orthogonality of the cosine basis functions, simple gradient algorithms, such as the Normalized Least Mean Squares (NLMS), can benefit from it and present a controlled and predictable convergence time and error misadjustment. Due to its simplicity, the proposed technique ranks as the best in terms of learning quality versus complexity, and it is presented as an attractive technique to be used in more complex supervised learning systems. Simulations illustrate the performance of the approach. This paper celebrates the 50th anniversary of the publication of the DCT by Nasir Ahmed in 1973.

翻译：本文研究以余弦函数作为基函数，对无记忆单变量连续函数进行逼近。本工作采用监督学习获取逼近系数，而非直接使用离散余弦变换(DCT)。由于余弦基函数的有限动态范围与正交性，简单梯度算法（如归一化最小均方算法(NLMS)）可从中获益，并呈现出可控、可预测的收敛时间与误差失调。因其简洁性，所提方法在学习质量与复杂度权衡中表现最优，成为适用于更复杂监督学习系统的颇具吸引力的技术。仿真实验验证了该方法的性能。本文谨以此纪念纳西尔·艾哈迈德(Nasir Ahmed)于1973年发表DCT五十周年。