Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured

Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2

On arbitrary polygonal domains $Omega subset RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s in (2,frac{5}{2})$ to $s in (1,frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned

Multiwavelets based on hermite cubic splines

the convection-diffusion equation. We use an implicit scheme for the time discretization an

Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability

USING SPLINE FUNCTIONS FOR THE SUBSTANTIATION OF TAX POLICIES BY LOCAL AUTHORITIES

The paper aims to approach innovative financial instruments for the management of public resources. In the category of these innovative tools have been included polynomial spline functions used for budgetary sizing in the substantiating of fiscal and budgetary policies. In order to use polynomial spline functions there have been made a number of steps consisted in the establishment of nodes, the calculation of specific coefficients corresponding to the spline functions, development and determination of errors of approximation. Also in this paper was done extrapolation of series of property tax data using polynomial spline functions of order I. For spline impelementation were taken two series of data, one reffering to property tax as a resultative variable and the second one reffering to building tax, resulting a correlation indicator R=0,95. Moreover the calculation of spline functions are easy to solve and due to small errors of approximation have a great power of predictibility, much better than using ordinary least squares method. In order to realise the research there have been used as methods of research several steps, namely observation, series of data construction and processing the data with spline functions. The data construction is a daily series gathered from the budget account, reffering to building tax and property tax. The added value of this paper is given by the possibility of avoiding deficits by using spline functions as innovative instruments in the publlic finance, the original contribution is made by the average of splines resulted from the series of data. The research results lead to conclusion that the polynomial spline functions are recommended to form the elaboration of fiscal and budgetary policies, due to relatively small errors obtained in the extrapolation of economic processes and phenomena. Future research directions are taking in consideration to study the polynomial spline functions of second-order, third-order, Hermite spline and cubic splines of class C2 .fiscal policy, budget deficits, spline functions, budget justification, debt crisis

Option Pricing under Multifactor Black-Scholes Model Using Orthogonal Spline Wavelets

The paper focuses on pricing European-style options on several underlying assets under the Black-Scholes model represented by a nonstationary partial differential equation. The proposed method combines the Galerkin method with $L^2$-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with Rannacher time-stepping. To this end, we construct an orthogonal cubic spline wavelet basis on the interval satisfying homogeneous Dirichlet boundary conditions and design a wavelet basis on the unit cube using the sparse tensor product. The method brings the following advantages. First, the number of basis functions is significantly smaller than for the full grid, which makes it possible to overcome the so-called curse of dimensionality. Second, some matrices involved in the computation are identity matrices, which significantly simplifies and streamlines the algorithm, especially in higher dimensions. Further, we prove that discretization matrices have uniformly bounded condition numbers, even without preconditioning, and that the condition numbers do not depend on the dimension of the problem. Due to the use of cubic spline wavelets, the method is higher-order convergent. Numerical experiments are presented for options on the geometric average.Comment: 43 pages, 10 figure

Wavelet B-Splines Bases on the Interval for Solving Boundary Value Problems

The use of multiresolution techniques and wavelets has become increa-singly popular in the development of numerical schemes for the solution of differential equations. Wavelet’s properties make them useful for developing hierarchical solutions to many engineering problems. They are well localized, oscillatory functions which provide a basis of the space of functions on the real line. We show the construction of derivative-orthogonal B-spline wavelets on the interval which have simple structure and provide sparse and well-conditioned matrices when they are used for solving differential equations with the wavelet-Galerkin method.Facultad de Ingenierí