As a result of a rather long-time research started in 2016, this theory whose structure is based on a fixed variable and an algebraic inequality, improves and somehow generalizes the well-known least squares theory. In fact, the fixed variable has a fundamental role in constituting the least p-variances theory. In this sense, some new concepts such as p-covariances with respect to a fixed variable, p-correlation coefficient with respect to a fixed variable and p-uncorrelatedness with respect to a fixed variable are first defined in order to establish least p-variance approximations. Then, we obtain a specific system called p-covariances linear system and apply the p-uncorrelatedness condition on its elements to find a general representation for p-uncorrelated variables. Afterwards, we apply the concept of p-uncorrelatedness for continuous functions particularly for polynomial sequences and find some new sequences such as a generic two-parameter hypergeometric polynomial of 4F3 type that satisfy such a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an approximation for p-variances, an improvement for the approximate solutions of over-determined systems and an improvement for the Bessel inequality and Parseval identity. Finally, we generalize the notion of least p-variance approximations based on several fixed orthogonal variables.

翻译：自2016年启动的长期研究结果表明，基于固定变量与代数不等式构建的该理论，改进并在一定程度上推广了著名的最小二乘理论。事实上，固定变量在构建最小p方差理论中具有根本性作用。为此，我们首先定义了若干新概念，包括关于固定变量的p协方差、关于固定变量的p相关系数以及关于固定变量的p不相关性，以建立最小p方差逼近。随后，我们推导出称为p协方差线性系统的特定方程组，并对其元素施加p不相关性条件，从而获得p不相关变量的一般表示。接着，我们将p不相关概念应用于连续函数，特别是多项式序列，并发现满足此p不相关性的新序列，例如4F3型通用双参数超几何多项式。进一步地，我们得到了1协方差的上界、p方差的逼近、超定方程组近似解的改进、贝塞尔不等式与帕塞瓦尔等式的推广。最后，我们将基于多个固定正交变量的最小p方差逼近概念进行了泛化。