Trigonometric Fmn-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 m-transform technique for functions with one-variable in order to improve its approximation properties at the end points of [a,b] and then generalize the extended trigonometric Fm -transform technique to functions with more variables. The approximation and convergence properties of the direct and inverse multi-variable extended trigonometric Fm -transforms are discussed. The applicability of multi-variable trigonometric F m -transforms to approximate multi-variable functions are illustrated by some examples. Moreover, some direct formulas for the multi-variable extended trigonometric Fm -transforms of partial derivatives of multi-variable functions are obtained and they are applied to solving the Cauchy problem of the transport equation. Also, the application of multi-variable extended trigonometric Fm -transforms for image compression is described. Some examples for the validity of the obtained results about the partial differential equations and image compression are given. The results are compared with some existence ones in the literature.