Saturation and an iterative construction process

AbstractLet {j}Kj=1∞ be a sequence of kernels, let {aj}j=1∞ be a sequence of positive numbers, and let f0 be a measurable function. Setting J0 = f0, we study the convergence in Lp(1 ⩽ p ⩽ 2 and p = ∞) of the sequence of singular integrals {Jn}n = 1∞ defined inductively by Jn(x) = (an(2π)12∝−∞∞Jn−1(x − t)nK(ant)dt, x ϵ R.The convergence of {Jn} in L∞ finds an application in Bray-Mandelbrojt's “repeated averaging” construction concerning a non quasi-analytic class of functions