This book provides a rigorous examination of modern operator theory focusing on the construction and convergence of linear positive operators in generalized function spaces. In addition, the book explores how to create "mathematical shortcuts" that accurately represent complex, real-world data using simpler, more manageable formulas. By integrating advanced summability methods-such as A-statistical and deferred weighted convergence, the book provides refined techniques for function approximation. From Fourier signal processing to black-box optimization, these chapters contribute to bridging the gap between abstract functional analysis and its applications. Readers will gain a better understanding of modern linear positive operators and advanced summability methods that extend beyond the limitations of classical approximation theory. A versatile toolset for analyzing convergence within complex, generalized function spaces is provided and equips readers to deal with new problems in signal processing, black-box optimization, and modeling.

B.B. Jena, Ph.D., is an Assistant Professor in the Department of Mathematics at Sri Sri University, Cuttack, India. Dr. Jena has extensive research experience in the approximation theory of Fourier series, particularly focusing on Lipschitz, Hölder, and Zygmund classes of functions. His research includes work on summability methods, convergence analysis, and error bounds. He has several publications in reputed journals addressing approximation using summability means and has applied analytical methods using MATLAB and C++. His work has also addressed the Gibbs phenomenon and its mitigation using double Fourier series.

Hemen Dutta, Ph.D., is a Professor in the Department of Mathematics at Gauhati University, Guwahati, India. He has published over 200 publications as research articles, proceedings papers, and book chapters and has also authored and edited several books published by leading publishers. He has organized many academic events and served as a speaker in various national and international events. He is on the editorial board of several SCI/SCIE and Scopus indexed journals and is the receipt of several research projects and grants from various funding agencies. His research interests include applied analysis and mathematical modeling.

Susanta Kumar Paikray, Ph.D., is a Professor in the Department of Mathematics at Veer Surendra Sai University of Technology, Burla, India. He has authored more than 100 scientific research articles in peer-reviewed national and international journals, and his current research interests include several areas of pure and applied mathematical sciences including summability theory, Fourier series, approximation theory, functional analysis, statistical convergence, operations research, and inventory optimization.