Discrete differential operators in multidimensional Haar wavelet spaces

We consider a class of discrete differential operators acting on multidimensional Haar wavelet basis with the aim of finding wavelet approximate solutions of partial differential problems. Although these operators depend on the interpolating method used for the Haar wavelets regularization and the scale dimension space, they can be easily used to define the space of approximate wavelet solutions. 1. Introduction. Advantage has extensively taken upon wavelets in order to analyze a number of different applications, such as image processing, signal detection, geo-physics, medicine, or turbulent flows [8, 9, 10, 13, 14, 15, 18]. More mathematically focussed differential equations and even nonlinear problems have also been studied with this cargo of possibilities that wavelets have brought out into scene [1, 2, 6, 16]. In this paper, we study a Cauchy problem with the PDE u t = Lu, where the unknown function u(t, x) ∈ L 2 ([0, 1]) for each t ∈ [0,T ], x ∈ [0, 1] d , d ∈ N, and L is a (nonneces