Simultaneous approximation of functions and their (\(\psi\) ,\(\beta\) )- derivatives by Fourier sums
The class \(L^{\psi}_{\beta}\) with \(\psi\) (k), \(k\in N\) a sequence and \(\beta\in R\) is defined to be the set of \(2\pi\)-periodic summable functions f for which \[ \sum^{\infty}_{k=1}\frac{1}{\psi (k)}\quad (a_ k\quad \cos (kx+\beta \frac{\pi}{2})+b_ k\quad \sin (kx+\beta \frac{\pi}{2})) \] (where \(a_ k\), \(b_ k\) are the Fourier coefficients of f) is the Fourier series of some summable function, denoted by \(f^{\psi}_{\beta}\) and called the (\(\psi\),\(\beta\)) derivative of f. The author gives various estimates for the functionals of form \[ \sum_{n,m}(f;x;\alpha)=\sum^{m}_{i=1}\alpha_ i\psi_ i(n)(f^{\psi_ i}_{\beta_ i}(x)-S_{n-1}(f^{\psi_ i}_{\beta_ i};x)), \] n,m\(\in N\) defined on \(L^{\psi}_{\beta}\), where the derivatives \(f^{\psi_ i}_{\beta_ i}\) are in some sense of ``less order'' as \(f^{\psi}_{\beta}\) and \(S_{n-1}\) stands for the (n-1)-th Fourier sum.
simultaneous trigonometric approximation
scientific article; zbMATH DE number 4163473