We study one-dimensional oscillator integrals which arise as Fourier-Stieltjes transforms of smooth, compactly supported measures on smooth curves in Euclidean spaces and determine their decay at infinity, provided the curves satisfy certain geometric conditions.
We consider the maximal function where and 0 < a < 1. We prove the global estimate , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.
A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, and α1,m,m²,... = 1/(2m).
Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a “weak ε-Kronecker set” if for every φ:E → there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not “too many”...
A sum of exponentials of the form , where the are distinct integers is called an idempotent trigonometric polynomial (because the convolution of with itself is ) or, simply, an idempotent. We show that for every and every set of the torus with there are idempotents concentrated on in the sense. More precisely, for each there is an explicitly calculated constant so that for each with and one can find an idempotent such that the ratio is greater than . This is in fact...