Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

Mogoi N. Evans *

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

Amos Otieno Wanjara

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

Samuel B. Apima

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

*Author to whom correspondence should be addressed.


Abstract

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

Keywords: Orthogonal polynomials, Sobolev spaces, norm-attainability, Hilbert space, weighted Sobolev spaces, Sturm-Liouville eigenvalue problem, convergence of orthogonal polynomials, reproducing kernel, Bessel’s inequality, Sobolev embedding, derivative properties of orthogonal polynomials, uniqueness of orthogonal polynomials, compactness of embeddings, pointwise convergence, Riesz representation


How to Cite

Evans, M. N., Wanjara, A. O., & Apima, S. B. (2024). Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces. Asian Research Journal of Mathematics, 20(4), 1–7. https://doi.org/10.9734/arjom/2024/v20i4792

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