On the Riemann derivatives of \(C_ sP\)-integrable functions
The many results proved in this paper generalize work of \textit{P. L. Butzer} and \textit{W. Kozakiewicz} [Can. J. Math. 6, 572-581 (1954; Zbl 0056.056)] by replacing the L-integral of that paper by the \(C_ sP\)- integral, s a positive integer. A typical result is: If f, g are \(C_ sP\)-integrable over every closed subinterval of (a,b), and if \(\frac{b- a}{2s}>h_ 1>h_ 2>...>h_ n>...>0,\) \(\lim_{n\to \infty}h_ n=0,\) and \((*)\quad \Delta^ s_{2h_ n}f(x)\geq (2h_ n)^ sg(x)\) a.e. in \((a+sh_ n,\quad b-sh_ n),\) \(n=1,2,...,\) then for some s-convex function \(\psi\), \(f=I^ s_{\alpha}g+\psi\) a.e. in (a,b). (Here, \[ \Delta^ s_{2h}f(x)=\sum^{s}_{j=0}(-1)^ j\left( \begin{matrix} s\\ j\end{matrix} \right)f(x+(s-2j)h), \] and \[ I^ s_{\alpha}g(x)=\int^{x}_{\alpha}...\int^{t_ 1}_{\alpha}g(t)dt dt_ 1...dt_{s-1},\quad a<\alpha <b. \] The integrals are, from left to right, in the \(C_ 1P-,...,C_ sP\)-sense, respectively.) The results are then applied to trigonometric series. In particular, the above quoted theorem is used to prove: If a trigonometric series is such that (i) its s-integrated series converges everywhere to a continuous function, f say, (ii) (*) holds a.e. in [0,2\(\pi\) ] for some \(2\pi\)-periodic \(C_ sP\)- integrable function g, then the trigonometric series is a \(C_{s-1}P\)- Fourier series (of the s-th Peano derivative of f).
scientific article; zbMATH DE number 4132483