Wavelet approximation of functions belonging to generalized Lipschitz class by extended Haar wavelet series and its application in the solution of Bessel differential equation using Haar operational matrix (Q2001762)

The authors define a generalized Haar wavelet, whose basis functions are supported on intervals of length \(1/\mu\), \(\mu \ge 1\) integer, instead of \([0,1]\). They also consider a generalization of Lipschitz continuity, where functions in the class Lip\(_\alpha^{(s)}[0,1)\) satisfy \[ |f(x) - f(x+t)| \le C s(t) |t|^\alpha, \quad x, x+t \in [0,1), \] with \(s(t) |t|^\alpha \to 0\) as \(t \to 0\). They generalize existing results on best approximation for Lipschitz continuous functions by Haar wavelets, and apply this to estimate the accuracy of the numerical solution of the differential equation for the 0th order Bessel function by wavelet methods.