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Equations

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length. An equation is written as two expressions, connected by an equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides. The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain one or more terms. For example, the equation has left-hand side , which has four terms, and right-hand side , consisting of just one term. The names of the variables suggest that x and y are unknowns, and that A, B, and C are parameters, but this is normally fixed by the context (in some contexts, y may be a parameter, or A, B, and C may be ordinary variables).

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Perturbed Fourier uniqueness and interpolation results in higher dimensions

Joao Pedro Gonçalves Ramos, Martin Peter Stoller

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Fourier uniqueness and interpolation in Euclidean space

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EPFL2022

Fourier non-uniqueness sets from totally real number fields

Martin Peter Stoller

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on R-n which vanish on the "component-wise square root" of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres root mS(n-1) for integers m >= 0 and, as m -> infinity, there are similar to c(K)m(n-1) many points on the m-th sphere for some explicit constant c(K), proportional to the square root of the discriminant of K. This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K, we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes "root Lambda" for general lattices Lambda subset of R-n. Using results about lattices in Lie groups of higher rank we prove that if n >= 2 and a certain group Gamma(Lambda) >= PSL2.(R)(n) is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n >= 5 and all real lambda >= 2, Fourier interpolation results for sequences of spheres root 2m/lambda Sn-1, where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincare type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

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