Decay of the Fourier Transform: Analytic and Geometric by Alex Iosevich

By Alex Iosevich

The Plancherel formulation says that the L^2 norm of the functionality is the same as the L^2 norm of its Fourier remodel. this means that no less than on typical, the Fourier remodel of an L^2 functionality decays at infinity. This e-book is devoted to the research of the speed of this decay below numerous assumptions and situations, a ways past the unique L^2 surroundings. Analytic and geometric houses of the underlying capabilities engage in a unbroken symbiosis which underlines the wide variety affects and purposes of the thoughts below consideration.​

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The sum being estimated is transformed to the form ∂ mλ N −2m m+2≤dist(k,∂(N S))≤CN k N + σ1 uN1 + · · · + σm uNm ∂u1 · · · ∂um 2 . Here 0 < σj < 1, j = 1, . . , m. We can take any m such that d+1 ≤ m ≤ d. 2 d−1 If 2 is an integer the derivative is bounded, so this quantity is not greater d−1 than CN d−2−2 2 = O(1). Otherwise, estimating the derivative in the direction (σ1 , . . 46) 2( d−1 2 −m) , 44 Chapter 2. Oscillatory Integrals where 0 ≤ σm,k ≤ m and the sums are over m + 2 ≤ dist(k, ∂(N S)) ≤ CN .

Applications 39 approximation properties of f. It is natural to define such a family as a sequence of dilations of a fixed set D. This has been extensively studied when D is the cube (cubic case) D = {x ∈ Rd : |xj | ≤ 1, j = 1, 2, . . , d}, or the ball (spherical case) 1 D = {x ∈ Rd : |x| = (x21 + x22 + · · · + x2d ) 2 ≤ 1}. Their N -dilations are RD = {x ∈ Rd : |xj | ≤ N, j = 1, 2, . . , d} and RD = {x ∈ Rd : |x| ≤ N }, respectively. The other example of a family of sets is the family of rectangles {x ∈ Rd : |xj | ≤ Nj , Nj > 0, j = 1, 2, .

Ud−1 on S so that for θ0 = θ¯0 we d−1 have ξ = − j=1 u2j . Therefore d−1 ξ = −θ¯d−1 (θ0 − θ¯0 ) + ξ|θ0 =θ¯0 = z − u2j . ,d−1 ∂ 2 g(¯ x) ∂xj ∂xk det 1 2 . ,d−1 Since the Lebesgue measure dω on θ is given by d dω = i=1 θ¯i2 |θ¯d | 1 2 dx1 · · · dxd−1 , one can write (cf. 15)): α d−1 R(f ; θ0 , θ¯1 , . . , θ¯d ) = z− d−1 i=1 ϕ1 (u, z)du1 · · · dud−1 , u2i i=1 u2i ≤z where z = θ¯n−1 (θ0 − θ¯0 ) on θ, and d x)|J|− 2 2 ϕ1 (0, 0) = ϕ0 (¯ 1 d−1 2 i=1 θ¯i2 |θ¯d | 1 2 . 32 Chapter 2. Oscillatory Integrals Note that the first and the second quadratic forms, G and Q, of S at the point x¯ are given, respectively, by the matrices δij + ¯d−1 ) ∂g(¯ x1 , .

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