Least p-Variances Theory

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.