Restriction for homogeneous polynomial surfaces in $\mathbb {R}^3$

Double Hilbert Transforms along Polynomial Surfaces in R3

where P(s, t) is a polynomial in s and t with P(0,0)= 0, and ∇P(0,0)= 0. We call H the (local) double Hilbert transform along the surface (s, t,P (s, t)). The operator may be precisely defined for a Schwartz function f by integrating where ≤ |s| ≤ 1 and η ≤ |t | ≤ 1, and then taking the limit as ,η→ 0. The corresponding 1-parameter problem has been extensively studied (see [RS1], [RS2], and [S]...

Fourier Restriction Estimates to Mixed Homogeneous Surfaces

Let a, b be real numbers such that 2 ≤ a < b, and let φ : R → R a mixed homogeneous function. We consider polynomial functions φ and also functions of the type φ (x1, x2) = A |x1| + B |x2| . Let Σ = {(x, φ (x)) : x ∈ B} with the Lebesgue induced measure. For f ∈ S ( R ) and x ∈ B, let (Rf) (x, φ (x)) = f̂ (x, φ (x)) , where f̂ denotes the usual Fourier transform. For a large class of functions φ ...

Polynomial Interpolation in R3

K e y w o r d s L a g r a n g e interpolation, Simultaneous approximation, Freud weights 1. I N T R O D U C T I O N In this paper , we are concerned with construct ing in terpola t ing subspaces of polynomials of several variables of relat ively small dimension as well as the corresponding in terpola t ion formulae a l a Lagrange. DEFINITION 1. A (linear) subspace G C C(Rn) is called k-interpol...

On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2kZ. More precisely, given a Boolean function F : {0, 1}n → {0, 1}, define its k-lift to be Fk : {0, 1}n → {0, 2k−1} by Fk(x) = 2k−F(x) (mod 2k). We consider the fractional agreement (which we refer to as γd,k(F)) of Fk with degree d polynomials from Z/2 Z[x1, . . . , xn]. Our results are the following: • Incr...

The Conway Polynomial in R3 and in Thickened Surfaces: A New Determinant Formulation