An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions

Mohammad Masjed-Jamei ()
Additional contact information
Mohammad Masjed-Jamei: Faculty of Mathematics, K. N. Toosi University of Technology, Tehran P.O. Box 16315-1618, Iran

Mathematics, 2025, vol. 13, issue 14, 1-89

Abstract: We establish a theory whose structure is based on a fixed variable and an algebraic inequality and which improves the well-known least squares theory. The mentioned fixed variable plays a basic role in creating such a theory. In this direction, some new concepts, such as p-covariances with respect to a fixed variable, p-correlation coefficients with respect to a fixed variable, and p-uncorrelatedness with respect to a fixed variable, are defined in order to establish least p-variance approximations. We then obtain a specific system, called the 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 we find some new sequences, such as a generic two-parameter hypergeometric polynomial of the F 3 4 type that satisfies a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an improvement to the approximate solutions of over-determined systems and an improvement to the Bessel inequality and Parseval identity. Finally, we generalize the concept of least p-variance approximations based on several fixed orthogonal variables.

Keywords: Least p-Variance approximations; least squares theory; p-Covariances and p-Correlation coefficients; p-Uncorrelatedness with respect to a fixed variable; Hypergeometric polynomials; generalized Gram-Schmidt orthogonalization process (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/14/2255/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/14/2255/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:14:p:2255-:d:1700011

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().