The Study of Applications of Real Analysis in Approximation Theory and Numerical Analysis
DOI:
https://doi.org/10.31305/rrijm2023.v03.n04.007Keywords:
Real analysis, Approximation theory, Numerical analysis, Convergence and stabilityAbstract
Real analysis provides the rigorous foundation for understanding fundamental concepts such as limits, continuity, differentiability, sequences, and series, all of which are critical for applied mathematical disciplines. This study explores the integration of real analysis principles into approximation theory and numerical analysis, emphasizing how theoretical insights guide the development of accurate, stable, and efficient computational methods. By analyzing convergence theorems, uniform approximation, and function continuity, the paper demonstrates how real analysis ensures predictable and reliable behavior of numerical algorithms, including iterative methods, interpolation techniques, and numerical integration schemes. Additionally, the study examines the error estimation and stability criteria that are derived from real analysis, highlighting their importance in evaluating the performance of approximation methods. Case studies and practical examples illustrate the direct application of theoretical concepts to computational problem-solving, bridging the gap between abstract mathematics and real world computations. The findings underscore that a strong conceptual understanding of real analysis is indispensable for researchers and practitioners, providing both theoretical rigor and practical utility in solving complex problems in approximation theory, numerical methods, and applied mathematics. This work ultimately contributes to a comprehensive understanding of the interplay between pure mathematical theory and computational practice, offering valuable insights for advancing numerical techniques and improving the reliability of approximations in various applied contexts.
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