F-transform of multi-variable functions, Trigonometric $F^m$-transform, Fuzzy partitions; Basic functions; Approximation; Partial differential equations; Transport equation; Image processing

In this study, we focus on the extention of the trigonometric $F$-transform for functions in one variable to: (i) a larger domain; (ii) a higher degree of the $F^m$-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 $F^{m}$-transforms are discussed. Then direct formulas for partial derivatives of functions of several variables are obtained in terms of trigonometric $F^{m}$-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, JPEG-FT. We have shown that this $^t\bar{F}^ {11 }$-transform image compression method has high accuracy and reasonably low (irredicible) complexity.