Trigonometric Fmn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{mn}$$\end{document}-transform of multi-variable functions and its application to the partial differential equations and image processing

In this study, we focus on the extension of the trigonometric F-transform for functions in one variable to: (i) a larger domain, (ii) a higher degree of the Fm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^m$$\end{document}-transform, and (iii) many-variable functions to improve its approximation properties over the entire domain and especially at its boundaries. In addition, the properties of approximation and convergence of direct and inverse extended and multidimensional trigonometric Fm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{m}$$\end{document}-transforms are discussed. Then, direct formulas for partial derivatives of functions of several variables are obtained in terms of trigonometric Fm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{m}$$\end{document}-transforms, which are used to solve the Cauchy problem for the transport equation. A new image compression method is proposed and compared with well-established compression methods such as JPEG, JPEG 2000 and their less complex variations JPEG(APDCBT), JPEG(APUBT3), APUBT3-NUP, and JPEG-FT. We have shown that this tF¯11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^t\bar{F}^ {11 }$$\end{document}-transform image compression method has high accuracy and reasonably low (irreducible) complexity.