Behavior of the group of derivations on sets of (\(\psi\) ,\(\beta\) )- differentiable functions (Q752387)

Let \(f\in L(0,2\pi)\) be a \(2\pi\)-periodic function and \[ a_ 0/2+\sum^{\infty}_{k=1}(a_ k \cos kx+b_ k \sin kx) \] be its Fourier series. If \(\psi\) :\({\mathbb{N}}\to {\mathbb{R}}\), \(\beta\in {\mathbb{R}}\) and the series \[ \sum^{\infty}_{k=1}\frac{1}{\psi (k)}[a_ k \cos (kx+\beta \pi /2)+b_ k \sin (kx+\beta \pi /2)] \] appears to be the Fourier series of a certain function, say \(f^{\psi}_{\beta}\), then this function is called the (\(\psi\),\(\beta\))-derivative of the function f. The authors develop their previous results concerning the classification and approximation of periodic functions which possess (\(\psi\),\(\beta\))-derivatives.