Trigonometric $$F^m$$ F m -transform and its approximative properties

The paper is devoted to generalizing the F-transform with constant components to the trigonometric \(F^m\)-transform (or \(^{t}F^{m}\)-transform for short) where \(^{t}F^{m}\)-transform components are trigonometric polynomials up to m degree, \(m\ge 0\). The involving basic functions for \(^{t}F^{m}\)-transform are sinusoidal shaped functions which are smooth functions. Applying the Gram–Schmidt procedure in order to achieve an orthogonal system leads us to simple calculations of trigonometric basis functions, and we obtain an explicit representation for them for arbitrary m. The approximation and convergence properties of the direct and inverse \(^{t}F^{m}\)-transforms are discussed, and the applicability of \(^{t}F^{m}\)-transforms is illustrated by some examples.