| The purpose of this work is the localization and identification of three different type of singularities of a given function: (n, alpha)-type singularities, chirps and poles.;The first type, called (n, alpha)-type singularity, is exemplified with fx=x-a n&vbm0;x-a&vbm0;a,;where n∈N,a∈R and 0 < alpha ≤ 1 are unknown parameters.;We want to determine the location of the singularity, that is to say, the value a, we want to determine the biggest value k for which f ∈ Ck, so that k = n here, and finally we want to detect the value alpha, that is to say, we want to be able to assert that fn∈Ca and that fn&nisin;C b for any beta > alpha. Recall that f is in Calpha if there exist a constant K such that; &vbm0;fx-f y&vbm0;≤K&vbm0;x-y&vbm0;a ,x,y∈ In addition we want to determine the values of these parameters with good accuracy and at low cost.;Detection and estimation of these singularities occur in many problems of importance in applications, like economics, statistics, especially time series analysis, signal and image processing, geophysics and data compression.;This identification problem is adequately solved, both theoretically and computationally, via a modification of the classical Littlewood-Paley theory together with the construction of an appropriate wavelet-based band pass filter.;Our next detection problem pertains to chirps. A typical chirp is exemplified by the function f (x) = x2sin1x . There has been a lot of research activity on chirps recently because of their importance in digital signal processing. The last problem treated in this work is that of pole detection from boundary measurements. Here a meromorphic function f is known in some interval contained in the domain of analyticity of f. From this knowledge we try to localize the poles of the function f. The problem of pole detection from boundary measurement occurs routinely in non destructive testing.;Among the three types of singularities studied in this work---( n, alpha)-type singularities, chirps and poles---chirps are those the most difficult to detect and only very preliminary results in the detection of chirps have been obtained. |
