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Concept# Square root

Summary

In mathematics, a square root of a number x is a number y such that y^2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y \cdot y) is x. For example, 4 and −4 are square roots of 16 because 4^2 = (-4)^2 = 16.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by \sqrt{x}, where the symbol "\sqrt{~^~}" is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt{9} = 3. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative x, the principal square root can also be written in exponent notation, as x^{1/2}

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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.

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.

In every dimension d >= 2, we give an explicit formula that expresses the values of any Schwartz function on R-d only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square root of an integer. We thus generalize an interpolation theorem by Radchenko and Viazovska [Publ. Math. Inst. Hautes Etudes Sci. 129 (2019), pp. 51-81] to higher dimensions. We develop a general tool to translate Fourier uniqueness and interpolation results for radial functions in higher dimensions, to corresponding results for non-radial functions in a fixed dimension. In dimensions greater or equal to 5, we solve the radial problem using a construction closely related to classical Poincare series. In the remaining small dimensions, we combine this technique with a direct generalization of the Radchenko-Viazovska formula to higher-dimensional radial functions, which we deduce from general results by Bondarenko, Radchenko and Seip [Fourier interpolation with zeros of zeta and L-functions, arXiv:2005.02996, 2020]

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