An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions
Abstract
1. Introduction
- and
2. Least p-Variances Approximation Based on a Fixed Variable
2.1. Continuous Case of p-Covariances Linear System
2.2. Discrete Case of p-Covariances Linear System
2.3. Some Reducible Cases of the p-Covariances Linear System
2.3.1. First Case
2.3.2. Second Case
2.4. An Illustrative Example and the Role of the Fixed Variable in It
2.4.1. First Case
2.4.2. Second Case
2.5. How to Choose a Suitable Option for the Fixed Variable: A Geometric Interpretation
3. p-Uncorrelatedness Condition on the p-Covariances Linear System and Its Consequences
4. p-Uncorrelated Expansions with Respect to a Fixed Variable
4.1. A Biorthogonality Property
5. p-Uncorrelated Functions with Respect to a Fixed Function
5.1. Two Classifications for Deriving p-Uncorrelated Functions
5.1.1. First Classification
5.1.2. Second Classification
5.2. A Generalized Theorem with an Important Remark
5.2.1. An Extension of the Previous Theorem
5.2.2. Remark
- (ii) If , then and
- (iii) If , then and
6. Simple p-Uncorrelated Variables and Their Relationship with Orthogonal Sequences
6.1. A Biorthogonality Property
6.2. A Basic Remark
6.3. Simple Uncorrelated Functions of Classical Type
6.3.1. New Classes of Orthogonal Functions
First Sequence
- (i)
- First subsequence. For two given parameters , if
- To derive (142), we have used the identity
- (ii)
- Second subsequence. If are replaced in (137) as
- (iii)
- Third subsequence. If are replaced as
- (iv)
- Fourth subsequence. If , where are two free parameters, are replaced in (137) as
Second Sequence
7. p-Uncorrelated Polynomials Sequence (p-UPS)
7.1. A Generic Recurrence Relation for p-UPS
7.2. A Complete Uncorrelated Sequence of Hypergeometric Polynomials of Type
8. On the Ordinary Case of p-Covariances
9. A Class of Uncorrelated Polynomials Based on a Predetermined Orthogonal Polynomial
9.1. An Uncorrelated Sequence of Hypergeometric Polynomials of Type
Some Particular Trigonometric Cases
9.2. An Uncorrelated Sequence of Hypergeometric Polynomials of Type
10. A Unified Approach for the Polynomials Obtained in Section 7, Section 8 and Section 9
11. p-Uncorrelated Vectors with Respect to a Fixed Vector
- Also, the notions of p-covariance and p-variance can be defined for these two vectors (with respect to ) as follows:
12. An Upper Bound for 1-Covariances
13. An Improvement to the Approximate Solutions of Over-Determined Systems
14. An Improvement of the Bessel Inequality and Parseval Identity
14.1. First Type of Improvement
14.2. Second Type of Improvement
15. Least p-Variances with Respect to Fixed Orthogonal Variables
15.1. Least p-Variances Approximation Based on Fixed Orthogonal Variables
15.1.1. First Case
15.1.2. Second Case
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Symbol | Weight Function | Kind, Interval & Parameters Constraint |
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Infinite, | ||
- | ||
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Masjed-Jamei, M. An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions. Mathematics 2025, 13, 2255. https://doi.org/10.3390/math13142255
Masjed-Jamei M. An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions. Mathematics. 2025; 13(14):2255. https://doi.org/10.3390/math13142255
Chicago/Turabian StyleMasjed-Jamei, Mohammad. 2025. "An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions" Mathematics 13, no. 14: 2255. https://doi.org/10.3390/math13142255
APA StyleMasjed-Jamei, M. (2025). An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions. Mathematics, 13(14), 2255. https://doi.org/10.3390/math13142255