Square-integrable function | EPFL Graph Search

In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [a,b] for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class o

Inversion de Fourier ponctuelle

Emanuel Enrico Seifert

Let f be an integrable function on RN, a a point in RN and B a complex number. If the mean value of f on the sphere of centre a and radius r tends to B when r tends to 0, we show that the Fourier integral at a of f is summable to B in Cesàro means of order λ > (N-1)/2. Let now U be a bounded open subset of RN whose boundary ∂U is a real analytic submanifold of RN with dimension N-1. We deduce from the preceding result that the Fourier integral at a of the indicator function of U is summable in Cesàro means of order λ > (N-1)/2 to 1 if a ∈ U, to 1/2 if a ∈ ∂U and to 0 if a ∉ U. We then show that if the function defined on ∂U by y → ‖ y - a ‖ has only a finite number of critical points, then we can take λ less or equal to (N-1)/2 ; more precisely, it suffices to have λ > (N-3)/2 + σ(a|∂U), where σ (a|∂U) < 0 is the maximum of the oscillatory indices associated to the critical points of y → ‖ y - a ‖ ; this generalizes results obtained by Pinsky, Taylor and Popov in 1997. Finally, writing μ∂U for the natural measure supported by ∂U, P(D) for a differential operator with constant coefficients of order m and b for a C∞ function on RN, we show that, if a is a point outside ∂U such that ‖ y - a ‖ has only a finite number of critical points on ∂U, the Fourier integral at a of the distribution P(D) bμ∂U is summable to 0 in Cesàro means of order λ > (N-1)/2 + m + σ (a|∂U) ; this generalizes a result obtained by Gonzàlez Vieli in 2002.

EPFL2007

La détection de périodicités cachées

Jean-Marc Nicoletti

This thesis is a small part of the preparation of the launch of the Gaia mission, a satellite of the European Space Agency (ESA). One of the goals of the mission is to perform a classification among variable stars considering different attributes. Periodic behavior in the observed light curve is such an attribute. It is of importance to determine these hidden cycles with as high an accuracy as possible. A key difficulty is connected to the fact that observations are taken at irregularly distributed time points. Classical methods of frequency analysis do not work in this situation. In a first step, we made a catalogue of potential solutions and applied them to real and simulated data. The performances have been compared in order to select the best method. In a second step, we considered the asymptotic performance of estimators based on regression methods. We derived the asymptotic distribution under the hypothesis that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), with Gaussian correlated errors. The irregular sampling has to satisfy the asymptotic property that the whole interval [0, P0) can be observed. We have also considered the special case of a periodic signal with period P0 > 0, piecewise constant on [0; P0) and the asymptotic distribution of the estimator. If an irregular sampling scheme does not satisfy the asymptotic property that the signal can be fully observed, the period can still be estimated reliably. We determined the asymptotic distribution of an estimator in a particular situation and under the assumption that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), and is observed with additive errors that are independent and identically distributed.

EPFL2012

A Sampling Theory for Non-Decaying Signals

Quy Ha Nguyen, Michaël Unser

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted

L _{ p }

spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d∕p + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the

L _{ p }

sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

2017

A Sampling Theory for Non-Decaying Signals

Quy Ha Nguyen, Michaël Unser

L _{ p }

L _{ p }

sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

2017