Application of fast spherical Fourier transform to density estimation

This paper is on density estimation on the 2-sphere, S², using the orthogonal series estimator corresponding to spherical harmonics. In the standard approach of truncating the Fourier series of the empirical density, the Fourier transform is replaced with a version of the discrete fast spherical Fourier transform, as developed by Driscoll and Healy. The fast transform only applies to quantitative data on a regular grid. We will apply a kernel operator to the empirical density, to produce a function whose values at the vertices of such a grid will be the basis for the density estimation. The proposed estimation procedure also contains a deconvolution step, in order to reduce the bias introduced by the initial kernel operator. The main issue is to find necessary conditions on the involved discretization and the bandwidth of the kernel operator, to preserve the rate of convergence that can be achieved by the usual computationally intensive Fourier transform. Density estimation is considered in L²(S²) and more generally in Sobolev spaces H_v(S²), any v?0, with the regularity assumption that the probability density to be estimated belongs to H_s(S²) for some s>v. The proposed technique to estimate the Fourier transform of an unknown density keeps computing cost down to order O(n), where n denotes the sample size.     

Numerical evaluation of finite Fourier integrals

A computationally efficient algorithm for evaluating Fourier integrals ∫¹_?1?(x)e^iωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered.     

The Definition of the Fourier Transform for Weighted Inequalities

Weighted norm inequalities for the Fourier transform have a long history, motivated in part by generalizations of the differentiation formula for Fourier transforms and by applications such as Kolmogorov′s prediction theory and restriction theory. Typically, a norm inequality, || ||L^q_μ ≤ ||f||L^p_v, is established on a subspace of L¹ contained in the weighted space L^p_v, where is the Fourier transform and μ and v are weights. The problem of defining the extension of on all of L^p_v is posed and solved here for several important cases. This definition sometimes results in the ordinary distributional Fourier transform, but there are other situations which are also analyzed. Precise necessary conditions and closure theorems are inextricably related to this program, and these results are established.     

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes (α>1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R^k is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformations is established.     

On Fourier Transforms of Radial Functions and Distributions

We find a formula that relates the Fourier transform of a radial function on R ⁿ with the Fourier transform of the same function defined on R ⁿ⁺². This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t?f(|t|) and the two-dimensional function (x ₁,x ₂)?f(|(x ₁,x ₂)|). We prove analogous results for radial tempered distributions.     

On Radial and Conical Fourier Multipliers

We investigate connections between radial Fourier multipliers on ℝ ^d and certain conical Fourier multipliers on ℝ ^d+1. As an application we obtain a new weak type endpoint bound for the Bochner–Riesz multipliers associated with the light cone in ℝ ^d+1, where d≥4, and results on characterizations of L ^p →L ^p,ν inequalities for convolutions with radial kernels.     

Bilinear Fourier Restriction Theorems

We provide a general scheme for proving L ^p estimates for certain bilinear Fourier restrictions outside the local L ² setting. As an application, we show how such estimates follow for the lacunary polygon. In contrast with prior approaches, our argument avoids any use of the Rubio de Francia Littlewood?CPaley inequality.     

On Fourier frame of absolutely continuous measures

Let μ be a compactly supported absolutely continuous probability measure on Rn, we show that L²(K,dμ) admits a Fourier frame if and only if its Radon-Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R¹ and L²(K,dμ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<1, the L² space of the λ-Bernoulli convolutions cannot admit a Fourier frame.     

Time-frequency localization for the short time Fourier transform

We use some estimating of orthogonal projection in a reproducing kernel Hilbert space, to prove a sharp quantitaive form of Shapiro's mean dispersion theroem with generalized dispersion for the short time Fourier transform. Other forms of localization of orthonormal sequences in L²?^d) notably the umbrella theorem, are also proved for the short time Fourier transform.     

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p‐computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p‐computable Baire categories. We show that L^p‐computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is L^p‐computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p‐computable functions is not meager. We give an alternative characterization of meager sets via Banach‐Mazur games. We study the convergence of Fourier series for L^p‐computable functions and show that whereas for every p > 1, the Fourier series of every L^p‐computable function f converges to f in the L^p norm, the set of L¹‐computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Convolutions for the Fourier transforms with geometric variables and applications

This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier‐cosine, Fourier‐sine transforms. With respect to applications, by using the constructed convolutions normed rings on L₁( R ⁿ) are constructed, and explicit solutions of integral equations of convolution type are obtained (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of Fourier multipliers on starlike Lipschitz surfaces

In this paper, we consider a class of Fourier multipliers whose symbols are controlled by a polynomial on starlike Lipschitz surfaces and get the L² boundedness of these operators on Sobolev spaces and their endpoint estimates.     

Weighted integrability of multiplicative Fourier transforms

We study the relationship between the weighted integrability of a function and that of its multiplicative Fourier transform (MFT). In particular, for the MFT we prove an analog of R. Boas’ conjecture related to the Fourier sine and cosine transforms. In addition, we obtain a sufficient condition under which a contraction of an MFT is also an MFT. For the moduli of continuity ω satisfying N.K. Bari’s condition, we present a criterion for determining whether a function with a nonnegative MFT belongs to the class H ^ω .     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

Localisation and weighted inequalities for spherical Fourier means

In this work, we establish certain equivalences between the localisation properties with respect to spherical Fourier means of the support of a given Borel measure and the L ²-rate of decay of the Fourier extension operator associated to it. This, in turn, is intimately connected with the property that the X-ray transform of the measure be uniformly bounded. Geometric properties of sets supporting such a measure are studied. The research of this paper has been partially supported by the EU Comission via the network HARP, and by MEC Grant MTM2004-00678. The first author was partially supported by a Leverhulme Study Abroad Fellowship.     

Qualitative uncertainty principles for the hypergeometric Fourier transform

We prove an L ^p version of the Donoho–Stark’s uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\). Next, using the ultracontractive properties of the semigroups generated by the Heckman–Opdam Laplacian operator, we obtain an L ^p Heisenberg–Pauli–Weyl uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\).