A regularity class for the roots of nonnegative functions

Abstract

We investigate the regularity of the positive roots of a nonnegative function of one-variable. A modified Hölder space F β is introduced such that if f ∈ F β then f α ∈ C αβ . This provides sufficient conditions to overcome the usual limitation in the square root case (α = 1/2) for Hölder functions that f 1/2 need be no more than C 1 in general. We also derive bounds on the wavelet coefficients of f α , which provide a finer understanding of its local regularity.

Minimax theory for a class of nonlinear statistical inverse problems

“…For density estimation, local rates of the type (3.5) have been obtained only up to β = 2 [31]. This is consistent with our results since H β and the space of all functions on [0, 1] that can be extended to a non-negative β-Hölder function on R in fact coincide for 0 < β ≤ 2 (Theorem 2.4 of [32]). The authors in [31] deal with higher derivatives in one specific situation, namely points near the support boundary, where the function necessarily satisfies a flatness type condition allowing one to quantify the behaviour of the higher order derivatives.…”

“…The presence of two regimes is caused by the non-linearity of h, which is smooth away from 0 but has irregular point 0. Indeed, the smoothness of h • f bears much less resemblance to that of f near 0 (see Lemma 5.3 of [32]).…”

“…so that again taking η > 0 small enough, we have |Rh β n K((·−x 0 )/h n )| H β ≤ (R−lim sup n f * n H β )/2. Using the definition of · H β and that it defines a norm by Theorem 2.1 of [32],…”

The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

“…Notice that H β (R) = C β (R) for β ≤ 1. Properties of the function space H β (R) are studied in [30].…”

“…Lemma 6 (Lemma 1 in [30]). Suppose that f ∈ H β with β > 0 and let a = a(β) > 0 be any constant satisfying (e a − 1) + a β /(⌊β⌋!)…”