Generalized 2-Microlocal Frontier Prescription
- PDF / 600,647 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 61 Downloads / 180 Views
Generalized 2-Microlocal Frontier Prescription Ursula Molter1 · Mariel Rosenblatt2 Received: 3 January 2020 / Revised: 21 August 2020 / Published online: 20 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in R2 , provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point x0 is a given line. Further, for a general concave downward curve, we obtain a large family of functions (or distributions) for which the 2-microlocal frontier is the given curve. This family contains—as special cases— the constructions given in Meyer (CRM monograph series, 1998), Guiheneuf et al. (ACHA 5(4):487–492, 1998) and Lévy Véhel et al. (Proc Symp Pure Math 72:319– 334, 2004). Moreover, following Lévy Véhel et al. (2004), we extend our results to the prescription on a countable dense set. Keywords 2-Microlocal analysis · Pointwise regularity · Regularity exponents · Wavelet analysis Mathematics Subject Classification 25A16 · 42C40 · 60G35
1 Introduction The detection of the local roughness of mathematical objects, such as functions, measures, stochastic processes or fractal sets, has gained a relevant place in Mathematics
Communicated by Stephane Jaffard.
B
Ursula Molter [email protected] Mariel Rosenblatt [email protected]
1
Dto. de Matemática, FCEyN and IMAS, Universidad de Buenos Aires and CONICET, Ciudad Autónoma de Buenos Aires, Argentina
2
IDH-ICI, Universidad Nacional de General Sarmiento, Los Polvorines, Buenos Aires, Argentina
88
Page 2 of 28
Journal of Fourier Analysis and Applications (2020) 26:88
over the past decades. One key goal is to identify the most representative singularities of the object. Singularities of functions or signals are points at which the function lacks regularity. The identification and characterization of singularities are an important topic in signal processing, since they contain significant information about the phenomena. There are several types of singularities that can be illustrated in known examples, e.g. the function f (x) = |x − x0 |α , 0 < α < 1, has a cusp type singularity at x0 , which is non oscillating . In contrast, the functions f (x) = |x − x0 |α sin |x − x0 |−β , with 0 < α < 1 and β > 0, and f (x) = |x − x0 | sin |x − x0 |−1 + |x − x0 |3/2 have an oscillatory behaviour around x0 . Oscillating and non oscillating are a first and clear distinction among singularities. However, the intuitive notion of oscillation is not enough to characterize more complex structures. For example, in the examples above though both are oscillating the first one has a chirp type singularity whereas the second one not. From a mathematical point of view, it is important to characterize the different singularities. Classical fu
Data Loading...
