Hilbert transforms and maximal functions along variable flat curves

We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.

     

estimates for a class of oscillatory integrals

We obtain a simpler proof of Theorem 3.1 of The complete mapping properties of some oscillatory integrals in several dimensions, by G. Sampson and P. Szeptycki (Canad. Math. J. 53 (5) (2001), 1031-1056).

     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions

We evaluate explicitly the integrals , with the being any one of the four Chebyshev polynomials of degree . These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing in its interior.

     

Sharp maximal estimates for doubly oscillatory integrals

We study doubly oscillatory integrals

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .

     

Counterexamples to the well-posedness of transmission boundary value problems for the Laplacian

In this note we show that the well-posedness range for transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp. Our approach relies on Mellin transform techniques for singular integrals naturally associated with the transmission problems and on a careful analysis of the spectra of such singular integrals.

     

Multiple path-valued conditional Yeh-Wiener integrals

In this paper we establish various results involving parallel line-valued conditional Yeh-Wiener integrals of the type , where . We then develop a formula for converting these multiple path-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. Next, conditional Yeh-Wiener integrals for functionals of the form

are evaluated by solving an appropriate Wiener integral equation. Finally, a Cameron-Martin translation theorem is obtained for these multiple path-valued conditional Yeh-Wiener integrals.

     

Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

We consider the approximation of -dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals is . Since this is a worst case result, compared to the expected convergence rate of Monte Carlo, it shows the superiority of quasi-Monte Carlo for this type of integral. This is much faster than the worst case convergence, , of quasi-Monte Carlo.

     

Cubature formulas for symmetric measures in higher dimensions with few points

We study cubature formulas for -dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree or and large dimension the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.

We also show, for any odd degree , that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

     

On asymptotic approximations of the residual currents

We use a -module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents in the complete intersection case. Our results also provide asymptotic developments for these integrals.

     

Convergence of two-dimensional weighted integrals

A two-dimensional weighted integral in is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.

     

Singular integrals on symmetric spaces, II

We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by -bi-invariant kernels with certain large scale cancellation properties. As an application we prove boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.

     

Spaces between and

In this paper we consider the spaces that lie between and . We discuss their interpolation properties and the behavior of maximal functions and singular integrals acting on them.

     

A Monte Carlo algorithm for weighted integration over

We present and analyze a new randomized algorithm for numerical computation of weighted integrals over the unbounded domain . The algorithm and its desirable theoretical properties are derived based on certain stochastic assumptions about the integrands. It is easy to implement, enjoys convergence rate, and uses only standard random number generators. Numerical results are also included.

     

Poisson integrals associated to Dunkl operators for dihedral groups

In this paper we study the boundary behavior of Poisson integrals associated to Dunkl differential-difference operators for dihedral groups and the boundary integral representations for functions on the unit disc of annihilated by the Laplace operator corresponding to these differential-difference operators.

     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

Sharpness of the Korányi approach region

We prove a Littlewood-type theorem which shows the sharpness of the Korányi approach region for the boundary behavior of Poisson-Szegö integrals on the unit ball of . Our result is stronger than Hakim and Sibony (1983).

     

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

We consider the numerical solution of the stochastic partial differential equation , where is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise ( ) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise ( ) we show that no such improvements are possible.

     

bounds for oscillatory hyper-Hilbert transform along curves

We study the oscillatory hyper-Hilbert transform

(1)

along the curve , where are some real positive numbers. We prove that if , then is bounded on whenever . Furthermore, we also prove that is bounded on when . Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an boundedness result for some strongly parabolic singular integrals with rough kernels.