Mock Fourier series and transforms associated with certain Cantor measures

We extend the construction, originally due to Jorgensen and Pedersen, of spectral pairs {μ, Λ}, consisting of Cantor measures μ on ℝⁿ and discrete sets Λ such that the exponentials with frequency in Λ form an orthonormal basis forL ²(μ). We give conditions under which these mock Fourier series expansions ofL ¹(μ) functions converge in a weak sense, and for a dense set of continuous functions the convergence is uniform. We show how to construct spectral pairs (²(μ) of infinite Cantor measures with unbounded support such that defined for a dense subset ofL ²(μ), extends to an isometry fromL ²(μ) ontoL ²(μ'), a kind of mock Fourier transform. Our constructions do not require self-similarity, but only a compatible product structure for the pairs. We also give an analogue of the Shannon Sampling Theorem to reconstruct a function whose Fourier transform is supported in the Cantor set associated with μ from its values on Λ. In memory of Irving Segal Research supported in part by the National Science Foundation, grant DMS 9970337.     

A certain class of Jensen measures for uniform algebras

For any uniform algebraA and any pointq of the maximal ideal space ofA there exists a Jensen measureλ forq carried on the Shilov boundary forA such thatλ admits the generalized Brownian maximal function to each nonnegativeA-subharmonic function inC _R(X). The maximal function and its original function satisfy Doob’s inequality, Burkholder-Gundy-Silverstein inequalities and Fefferman-Stein inequality.     

Singular measures andl p Fourier transforms

LetG be the Cantor group or the circle group, and let Γ be the dual ofG. There exists a probability measure λ onG, singular with respect to Haar measure onG, such that for 1≦p<2 the inequality {fx115-1} holds for trigonometric polynomialsf. Partially supported by NSF Grant MCS 76-02267-A01.     

Coin-tossing measures and their Fourier transforms

A coin-tossing measure μ on [0,1] is a probability measure satisfying

     

Fourier transforms and generalized Lipschitz classes in uniform metric

For functions f∈L¹(R)∩C(R) with Fourier transforms in L¹(R) we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes Hω,m and hω,m in terms of behavior of .     

Asymptotic behavior of Fourier transforms of self-similar measures

Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case

     

Norm inequalities for certain classes of functions and their Fourier transforms

We apply tools of interpolation theory and a commutative property of the Hilbert transform to prove necessary and sufficient conditions related to trigonometric series. These results extend and improve related theorems proven by several authors, summarized by Boas. In addition, we explore inequalities and operators, both connected to Hardy's inequalities, on certain classes of functions, including quasimonotone functions.     

Fourier transforms of irregular mixed homogeneous hypersurface measures

With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.     

Randomised circular means of Fourier transforms of measures

We explore decay estimates for circular means of the Fourier transform of a measure on in terms of its -dimensional energy. We find new upper bounds for the decay exponent. We also prove sharp estimates for a certain family of randomised versions of this problem.

     

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F (β)={t:lim n→∞‖tβ n‖= 0} is countable,where a is the distance between a real number a and the set of integers.In this paper it is proved that every member in this set is of the form cβ n,where ‖n‖ is a nonnegative integer and c is determined by a linear system of equations.Furthermore,for some self-similar measures μ associated with β,the limit at infinity of the Fourier transforms lim n→∞μ(tβ n)≠0 if and only if t is in a certain subset of F (β).This generalizes a similar result of Huang and Strichartz.     

Average decay estimates for Fourier transforms of measures supported on curves

We consider Fourier transforms of densities supported on curves in ℝ^d. We obtain sharp lower and close to sharp upper bounds for the decay rates of as R → ∞.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Fourier transforms of functions in Herz spaces on certain groups

G— - {G _n } _n =– , G _n =G G _n ={0}. G. K(,p,q;G) K(,p,q;) - G , . , G - (. . sup {order (G _n /G _n +1):=0, ± 1, ...<), K(.,p,q;G) L ^{p/(p–p–1),q} () L ^{p/(p–p–1),q} () K(-,p,q; ), 1<p2, 0<<1/p=1–1/p, 0<q. . . . .     

About the Fourier transforms of non-smooth measures on hyperspheres

If d_μ is the Fourier transform of a smooth measure d_μ on the hypersphere Sⁿ⁻¹ (n≥2) then there exists a constant C dependent only on n such that ⋎d_μ(y)⋎≤C(1+⋎y⋎)^−(n−1)/2 for all y∈Rⁿ. In this paper, we show that the above statement is false for non-smooth measures. And we present the corresponding estimations for the Fourier transforms of certain non-smooth measures on Sⁿ⁻¹. This research is supported by a grant of NSF of P. R. China.