Quantum symmetric 𝐿^{𝑝} derivatives

Abstract

Abstract. For 1 ≤ p ≤ ∞, a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For 1 ≤ p ≤ ∞, symmetrization holds, that is, whenever the L p kth Peano derivative exists at a point, all of these derivatives of order k also exist at that point. The main result, desymmetrization, is that conversely, for 1 ≤ p ≤ ∞, each L p symmetric quantum derivative is a.e. equivalent to the L p Peano derivative of the same order. For k = 1 and 2, each kth L p symmetric quantum derivative coincides with both corresponding kth L p Riemann symmetric quantum derivatives, so, in particular, for k = 1 and 2, both kth L p Riemann symmetric quantum derivatives are a.e. equivalent to the L p Peano derivative.

A generalization of the GGR conjecture

“…They have many applications in the theory of trigonometric series [36,39] and numerical analysis [10,28,35]. Quantum Riemann derivatives are studied in [3,9], and multidimensional Riemann derivatives are a part of [4]. For more on generalized Riemann differentiation see [32,33] and the survey article [2] by Ash.…”