The First Symmetric Derivative

Approximate high order smoothness

“…Since D~pf is continuous at xo, it is bounded on a neighborhood J of Xo and hence equals Dlf on J by part a) of this lemma. Applying the quasi-mean-value theorem of [1], we conclude that f is differentiable at x0.…”

The Symmetric Derivative

“…By assuming | ,4 | = 0 and following the same line of proof as above, we establish the first part of the second sentence in Theorem 7.1 ; the second part is established by considering -/. D Theorem 7.1 was apparently first proved by Aull [1] for continuous functions. It was later extended by Evans [5] and Kundu [10] to functions satisfying certain semicontinuity conditions.…”

“…Using/5 in place of/', various authors have established analogues of the common theorems of ordinary differential calculus, but most of their theorems have required the primitive functions to satisfy rather strong semicontinuity conditions. (See Aull [1], Evans [5], Kundu [10] or Weil [17].) In § §6 and 7 we will prove versions of these theorems for functions in the class w2.…”

Differentiation of Fourier Series via Orthogonal Derivative

“…This problem was remedied by Lanczos (1956) with ( ) defined as a Symmetric derivative (Aull 1967, Washburn 2006:…”