**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Fourier inversion theorem

Summary

In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the convention for the Fourier transform that
:(\mathcal{F}f)(\xi):=\int_{\mathbb{R}} e^{-2\pi iy\cdot\xi} , f(y),dy,
then
:f(x)=\int_{\mathbb{R}} e^{2\pi ix\cdot\xi} , (\mathcal{F}f)(\xi),d\xi.
In other words, the theorem says that
:f(x)=\iint_{\mathbb{R}^2} e^{2\pi i(x-y)\cdot\xi} , f(y),dy,d\xi.
This last equation is called the Fourier integral theorem.
Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then
:\mathcal{F}^{-1}=\mathca

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (5)

Related people

No results

Loading

Loading

Loading

Related concepts (10)

Related courses (24)

Related units

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric s

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g) th

MATH-205: Analysis IV

Learn the basis of Lebesgue integration and Fourier analysis

ENV-614: Fourier analysis and boundary value problems

Learning Fourier Series and Boundary Value Problems with a view to a variety of science and engineering problems. Learn the use of special functions like Bessel functions and applications. Introduce the doctoral students to general Sturm-Liouville problems and applications.

MATH-494: Topics in arithmetic geometry

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Zeta function.

No results

Related lectures (44)

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.

Thierry Blu, Djano Kandaswamy, Dimitri Nestor Alice Van De Ville

Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g. EEG) is the estimation of the locations and strengths of point sources from boundary data. We propose a new technique, for which we coin the term “analytic sensing”. First, generalized measures are obtained by applying Green's theorem to selected functions that are analytic in a given domain and at the same time localized to “sense” the sources. Second, we use the finite-rate-of-innovation framework to determine the locations of the sources. Hence, we construct a polynomial whose roots are the sources' locations. Finally, the strengths of the sources are found by solving a linear system of equations. Preliminary results, using synthetic data, demonstrate the feasibility of the proposed method.

Thierry Blu, Djano Kandaswamy, Dimitri Nestor Alice Van De Ville

Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g. EEG) is the estimation of the locations and strengths of point sources from boundary data.