A rational approximation in a triangle is proposed and analyzed in this paper. The rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform. Optimal error estimates for the L2− and H1 0−orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle. It is shown that the...

In this paper, for an operator defined by the action of an M -th order differential operator with polynomial-type coefficients on the function space L2 (k) (R) := {f : measurable|‖f‖(k) < ∞} with norm ‖f‖(k) = R |f(x)|2(x2 + 1)dx (k0 ∈ Z), we prove regularity (continuity and differentiability up to M times) of the eigenfunctions of its closure (with respect to the graph norm) under the conditio...

The technique introduced in this paper is a means for estimating and discovering underlying patterns for a large number of curves observed with heteroscedastic errors. Therefore, both the mean and the variance functions of each curve are assumed unknown and varying over time. The method consists of a series of steps. We transform using an orthonormal basis of functions in L2. In the transform d...

We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L2 to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by...

In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again for a fixed vector in the image space, ...

We study the general problem of oblique projections in discrete shift-invariant spaces of 12 and we give error bounds on the approximation. We define the concept of discrete multiresolutions and wavelet spaces and show that the oblique projections on certain subclasses of discrete multiresolutions and their associated wavelet spaces can be obtained using perfect reconstruction filter banks. The...

In this paper, a new class of Bayesian lower bounds is proposed. Derivation of the proposed class is performed via projection of each entry of the vector-function to be estimated on a closed Hilbert subspace of L2. This Hilbert subspace contains linear transformations of elements in the domain of an integral transform, applied on functions used for computation of bounds in the Weiss-Weinstein c...

Evaluation of nitronium ion-transfer equilibria, L1NO2+ + L2 = L2NO2+ + L1 (where L1 and L2 are ligands 1 and 2, respectively) by Fourier-transform ion cyclotron resonance mass spectrometry and application of the kinetic method, based on the metastable fragmentation of L1(NO2+)L2 nitronium ion-bound dimers led to a scale of relative gas-phase nitronium ion affinities. This scale, calibrated to ...

Wavelets are used in a wide range of applications such as signal analysis, signal compression, finite element methods, differential equations, and integral equations. In the following we will discuss the limitations of traditional basis expansions and show why wavelets are in many cases more efficient representations. A mathematical treatment of second generation wavelets as well as an example ...

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels jumps. Such arise as scaling limits of branching processes in population genetics self-exciting Hawkes mathematical finance. The strategy we adopt the existence part is based on approximations using L2-kernels combined a general result. Most importantly, establish weak duality argumen...