Filtered integration rules for finite weighted Hilbert transforms

Weighted Quadrature Rules for Finite Element Methods

We discuss the numerical integration of polynomials times exponential weighting functions arising from multiscale finite element computations. The new rules are more accurate than standard quadratures and are better suited to existing codes than formulas computed by symbolic integration. We test our approach in a multiscale finite element method for the 2D reaction-diffusion equation. Standard ...

Infinite-dimensional integration on weighted Hilbert spaces

We study the numerical integration problem for functions with infinitely many variables. The functions we want to integrate are from a reproducing kernel Hilbert space which is endowed with a weighted norm. We study the worst case ε-complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most ε. Here we assume that the c...

Two-dimensional Hilbert Transforms'

A Sufficient Condition for the Boundedness of Operator-weighted Martingale Transforms and Hilbert Transform

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space H. We show that if W and its inverse W−1 both satisfy a matrix reverse Hölder property introduced in [2], then the weighted Hilbert transform H : LW (R,H) → L 2 W (R,H) and also all weighted dyadic martingale transforms Tσ : LW (R,H)→ L 2 W (R,H) are bound...

Weighted Norm Inequalities for Hilbert Transforms and Conjugate Functions of Even and Odd Functions1

It is well known that the Hilbert tranformation and the conjugate function operator restricted to even (odd) functions define bounded linear operators on weighted If spaces under more general conditions than is the case for the unrestricted operators. In analogy with recent results of Hunt, Muckenhoupt and Wheeden for the Hilbert transform and the conjugate function operator, we obtain necessar...