Continuous singular measures with small Fourier-Stieltjes transforms

Representation of Functions as Absolutely Convergent Fourier-stieltjes Transforms

0. Introduction. Let G be a locally compact Abelian group, and G* its group of (continuous) characters. We write (y, x) to indicate the function on G*XG which is the value of the character y EG* at the point *6G, or, dually, the value of the character xEG at the point y EG*. Borel sets in G are taken to be the smallest cr-algebra of sets containing all closed sets. We consider the set of all re...

Fourier Series for Singular Measures

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [0, 1), every f ∈ L2(μ) possesses a Fourier series of the form f (x) = ∑n=0 cne. We show that the coefficients cn can be computed in terms of the quantities f̂ (n) = ∫ 1 0 f (x)e −2πinxdμ(x). We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ-bandlimited.

Surface Measures - Maximal Functions and Fourier Transforms

Continuous quaternion fourier and wavelet transforms

It is well-known that every two dimensional rotation around the origin in the plane R can be represented by the multiplication of the complex number e = cos θ + i sin θ, 0 ≤ θ < 2π. Similarly, every three dimensional rotation in the space R can be represented by the multiplications of the quaternion q from the left-hand side and its conjugate q̄ from the right-hand side, where q = cos(θ/2) + α s...

Fourier–bessel Functions of Singular Continuous Measures and Their Many Asymptotics *

We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier–Bessel functions, in the argument, the order, and in certain combinations of the two is required to solve a number of problems arising in quantum mechanics. We present known results, new approaches a...