Higher Order Derivatives

The concept of higher order derivatives is useful in many branches of mathematics and its applications. As they are useful in many places, nth order derivatives are often defined directly. Higher Order Derivatives discusses these derivatives, their uses, and the relations among them. It covers higher order generalized derivatives, including the Peano, d.l.V.P., and Abel derivatives; along with the symmetric and unsymmetric Riemann, Cesàro, Borel, LP-, and Laplace derivatives.

Although much work has been done on the Peano and de la Vallée Poussin derivatives, there is a large amount of work to be done on the other higher order derivatives as their properties remain often virtually unexplored. This book introduces newcomers interested in the field of higher order derivatives to the present state of knowledge. Basic advanced real analysis is the only required background, and, although the special Denjoy integral has been used, knowledge of the Lebesgue integral should suffice.

IntroductionHigher order derivativesDivided difference of order n General derivatives of order n Generalized Riemann derivatives of order nPeano derivativesRiemann* derivativesSymmetric de la Vallée Poussin derivativesSymmetric Riemann* derivativesCesàro derivatives Symmetric Cesàro derivativesBorel derivativesSymmetric Borel derivativesLp-derivativesSymmetric Lp-derivativesAbel derivativesLaplace derivativesSymmetric Laplace derivativesRelations between derivativesOrdinary and Peano derivatives Riemann*and Peano derivatives Symmetric Riemann* and symmetric de la Vallée Poussin derivativesCesàro and Peano derivativesPeano and symmetric de la Vallée Poussin derivatives and smoothness of order kSymmetric Cesàro and symmetric de la Vallée Poussin derivatives Borel and Peano derivativesSymmetric Borel and symmetric de la Vallée Poussin derivatives Borel and symmetric Borel derivatives and Borel smoothness of order k Peano and Lp-derivativesLp- and symmetric Lp-derivativesSymmetric de la Vallée Poussin and symmetric Lp-derivativesBorel and Lp-derivativesSymmetric Borel and symmetric Lp-derivativesCesàro and Borel derivativesSymmetric Cesàro and symmetric Borel derivativesAbel and symmetric de la Vallée Poussin derivatives Laplace, Peano and generalized Peano derivativesLaplace and Borel derivativesSymmetric Laplace and symmetric de la Vallée Poussin derivativesLaplace and symmetric Laplace derivativesPeano and the unsymmetric Riemann derivativesSymmetric de la Vallée Poussin and the symmetric Riemann derivativesGeneralized Riemann and Peano derivativesMZ- and Peano derivatives