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Concept# Fourier transform

Summary

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 transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into terms of the intensity of its constituent pitches.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting no

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In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space $\R^d.$ We confirm the intuition that as the bounded domain increases to the whole space, both solutions become arbitrarily close to one another. Both vanishing Dirichlet and Neumann boundary conditions are considered.We first study the nonlinear SHE in any space dimension with multiplicative correlated noise and bounded initial data. We prove that the solutions to SHE on an increasing sequence of domains converge exponentially fast to the solution to SHE on $\R^d.$ Uniform convergence on compact set is obtained for all $p$-moments. The conditions that need to be imposed on the noise are the same as those required to ensure existence of a random field solution. A Gronwall-type iteration argument is used together with uniform bounds on the solutions, which are surprisingly valid for the entire sequence of increasing domains.We then study SHE in space dimension $d\ge 2$ with additive white noise and bounded initial data. Even though both solutions need to be considered as distributions, their difference is proved to be smooth. If fact, the order of smoothness depends only on the regularity of the boundary of the increasing sequence of domains. We prove that the Fourier transform, in the sense of distributions, of the solution to SHE on $\R^d$ do not have any locally mean-square integrable representative. Therefore, convergence is studied in local versions of Sobolev spaces. Again, exponential rate is obtained.Finally, we study the Anderson model for SHE with correlated noise and initial data given by a measure. We obtain a special expression for the second moment of the difference of the solution on $\R^d$ with that on a bounded domain. The contribution of the initial condition is made explicit. For example, exponentially fast convergence on compact sets is obtained for any initial condition with polynomial growth. More interestingly, from a given convergence rate, we can decide whether some initial data is admissible.

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an applica-tion, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Via-zovska [1].(c) 2022 Published by Elsevier Inc.

We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only Schwartz function which, together with its Fourier transform, vanishes on these spheres, is the zero function. We show that this remains true if we replace the spheres by surfaces or discrete sets of points which are sufficiently small perturbations of these spheres. In a complementary, opposite direction, we construct infinite dimensional spaces of Fourier eigenfunctions vanishing on on certain discrete subsets of those spheres. The proofs combine harmonic analysis, the theory of modular forms and algebraic number theory.