An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions
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 t...
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2025-07-01
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| author | Mohammad Masjed-Jamei |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>F</mi><mn>3</mn><none></none><mprescripts></mprescripts><mn>4</mn><none></none></mmultiscripts></mrow></semantics></math></inline-formula> 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. |
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| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
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| spelling | doaj-art-6ea1d9ef56f248fda31ea8ecce09ad1e2025-08-20T03:36:14ZengMDPI AGMathematics2227-73902025-07-011314225510.3390/math13142255An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated FunctionsMohammad Masjed-Jamei0Faculty of Mathematics, K. N. Toosi University of Technology, Tehran P.O. Box 16315-1618, IranWe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>F</mi><mn>3</mn><none></none><mprescripts></mprescripts><mn>4</mn><none></none></mmultiscripts></mrow></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/13/14/2255Least p-Variance approximationsleast squares theoryp-Covariances and p-Correlation coefficientsp-Uncorrelatedness with respect to a fixed variableHypergeometric polynomialsgeneralized Gram-Schmidt orthogonalization process |
| spellingShingle | Mohammad Masjed-Jamei An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions Mathematics 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 |
| title | An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions |
| title_full | An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions |
| title_fullStr | An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions |
| title_full_unstemmed | An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions |
| title_short | An Improvement of Least Squares Theory: Theory of Least p-Variances Approximation and p-Uncorrelated Functions |
| title_sort | improvement of least squares theory theory of least p variances approximation and p uncorrelated functions |
| topic | 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 |
| url | https://www.mdpi.com/2227-7390/13/14/2255 |
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