Abel derivative and Abel continuity
The concept of an Abel derivative is given in the paper of \textit{S. Verblunsky} [Proc. Lond. Math. Soc., II. Ser. 34, 441--456 (1932; Zbl 0006.05302)], and the book of \textit{A. Zygmund} [Trigonometric series. Volumes I and II combined. With a foreword by Robert Fefferman. 3rd ed. Cambridge: Cambridge University Press (2002; Zbl 1084.42003)], but the first definition was that of the second order Abel derivative by \textit{S. J. Taylor} [Q. J. Math., Oxf. II. Ser. 6, 255--274 (1955; Zbl 0066.30002)]. Since then little was done on these derivatives although the second author did some work in [J. Lond. Math. Soc., II. Ser. 17, 87--96 (1978; Zbl 0385.42003); J. Lond. Math. Soc., II. Ser. 20, 319--326 (1979; Zbl 0423.42006)]. Finally, Abel derivatives of all orders were defined in a recent book, ``Higher order derivatives'', by the second author [Boca Raton, FL: CRC Press (2012; Zbl 1247.26004)]. If \(f(r,x) ={{1}\over{2}}a_0 + \sum_{n=1}^{\infty}(a_n\cos nx + b_n\sin nx)r^n\) then the upper and lower Abel derivates of \(f\) at \(x\) of order \(k\) are defined by \( \overline {AD}_kf(x)= \limsup_{r\to 1-}\frac{\partial^kf}{\partial x^k}(r,x),\;\underline {AD}_kf(x)= \liminf_{r\to 1-}\frac{\partial^kf}{\partial x^k}(r,x) \); in particular, the Abel derivative at \(x\) of order \(0\) is just the Abel sum at \(x\) of the Fourier series of \(f\). In the aforementioned book the basic properties of these derivatives were obtained and relations to the symmetric de la Vallée Poussin derivatives, also called the symmetric Peano derivatives, explored. The current paper takes up the detailed examination of properties of these derivatives, especially of the first order derivative which for Lebesgue integrable functions is a generalisation of the ordinary first symmetric derivative. In this case the derivative is shown to be a Baire-1 function and a monotonicity theorem is proved. Extensions of the results to the higher order derivatives present difficulties although the second author has proved, in the second reference above, that for Lebesgue integrable functions the existence of the finite \(k\)-th order symmetric de la Vallée Poussin derivative implies the existence of the \(k\)-th order Abel derivative with the same value.
scientific article; zbMATH DE number 6168745