Property A in an Infinite-Dimensional Space. I

   Abstract. We prove that an infinite-dimensional space of piecewise polynomial functions of degree at most n-1 with infinitely many simple knots, n ≥ 2 , satisfies Property A. Apart from its independent interest, this result allows us to solve an open classical problem (n ≥ 3 ) in theory of best approximation: the uniqueness of best L ₁ -approximation by n -convex functions to an integrable, continuous function defined on a bounded interval. In this first part of the paper we prove the case n=2 and give key results in order to complete the general proof in the second part.     

Convergence of CG and GMRES on a tridiagonal Toeplitz linear system

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax=b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for residuals of CG, MINRES, and GMRES on solving a tridiagonal Toeplitz linear system, where A is Hermitian or just normal. These expressions and bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first or second kind. AMS subject classification (2000)  65F10, 65N22     

The black-box fast multipole method

A new O(N)$O(N)$ fast multipole formulation is proposed for non-oscillatory kernels. This algorithm is applicable to kernels K(x,y)$K(x\text{,}y)$ which are only known numerically, that is their numerical value can be obtained for any (x,y)$(x\text{,}y)$. This is quite different from many fast multipole methods which depend on analytical expansions of the far-field behavior of K  , for |x-y|$|x-y|$ large. Other “black-box” or “kernel-independent” fast multipole methods have been devised. Our approach has the advantage of requiring a small pre-computation time even for very large systems, and uses the minimal number of coefficients to represent the far-field, for a given L²$L^2$ tolerance error in the approximation. This technique can be very useful for problems where the kernel is known analytically but is quite complicated, or for kernels which are defined purely numerically.     

Unwrapping magnetic resonance phase maps with Chebyshev polynomials

A phase-unwrapping algorithm, based on the method of moments, is introduced in this work. The proposed algorithm expands the phase map in terms of a two-dimensional Chebyshev series. The expansion coefficients are calculated by exploiting the orthogonality of Chebyshev polynomials of the first kind. The performance of the proposed phase-unwrapping algorithm is tested on a synthetic phase map and experimental phase maps of a uniform phantom, a human brain and a mouse torso, all acquired from 3-T magnetic resonance (MR) scanners. To impose additional burdens on the algorithm, we introduced magnetic field inhomogeneities to the phantom and human brain data by an external gradient coil. The proposed phase-unwrapping algorithm is found to perform well on the phantom data set in a low signal-to-noise ratio (SNR) environment and on the synthetic data set. The proposed algorithm is also found to perform well in in vivo data sets of the human brain and mouse torso. Results obtained from the in vivo MR data sets show that the proposed algorithm produced unwrapped phase maps that are nearly identical to those produced by a widely used phase-unwrapping algorithm, PRELUDE 2D in the fMRI Software Library.     

Pseudospectral analysis of the interfacial stability of free jets with different velocity profiles

A double fluid model for a liquid jet surrounded by a coaxial gas stream was constructed. The interfacial stability of the model was studied by Chebyshev pseudospectral method for different basic velocity profiles. The physical variables were mapped into computational space using a nonlinear coordinates transformation. The general eigenvalues of the dispersion relation obtained are solved by QZ method, and the basic characteristics and their dependence on the flow parameters are analyzed.     

Chebyshev approximation of the second kind of modified Bessel function of order zero

The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J. P. Boyd‘ s rational Chebyshev basis.     

An Accurate Solution of the Poisson Equation by the Finite Difference-Chebyshev-Tau Method

ＩｎｔｒｏｄｕｃｔｉｏｎＭａｎｙｓｃｈｏｌｏｒｓｈａｖｅｄｉｓｃｕｓｓｅｄｔｈｅｓｏｌｕｔｉｏｎｏｆｔｈｅｔｗｏ＿ｄｉｍｅｎｓｉｏｎａｌＰｏｉｓｓｏｎｅｑｕａｔｉｏｎ .ＡｍａｔｒｉｘｄｉａｇｏｎａｌｉｚａｔｉｏｎｍｅｔｈｏｄｗａｓｄｅｖｅｌｏｐｅｄｂｙＨａｉｄｖｏｇｅｌａｎｄＺａｎｇ[1]ｆｏｒｔｈｅｓｏｌｕｔｉｏｎｏｆｔｈｅｔｗｏ＿ｄｉｍｅｎｓｉｏｎａｌＰｏｉｓｓｏｎｅｑｕａｔｉｏｎ .Ｔｈｉｓｍｅｔｈｏｄｉｓｅｆｆｉｃｉｅｎｔｂｕｔｒｅｑｕｉｒｅｓａｐｒｅｐｒｏｃｅｓｓｉｎｇｃａｌｃｕｌａｔｉｏ…     

Sets with External Chebyshev Layer

Mathematical Notes -     

Alexandroff's Double Arrow Compact Space and Approximation Theory

A compact space Q similar to the compact space known as Alexandroff's double arrow space is constructed. It is shown that the real space C(Q) has no Chebyshev subspaces of codimension >1, but the complex space C(Q) has such subspaces.     

Error Analysis of Clenshaw's Algorithm for Evaluating Derivatives of a Polynomial

A forward rounding error analysis is presented for the extended Clenshaw algorithm due to Skrzipek for evaluating the derivatives of a polynomial expanded in terms of orthogonal polynomials. Reformulating in matrix notation the three-term recurrence relation satisfied by orthogonal polynomials facilitates the estimate of the rounding error for the m-th derivative, which is recursively estimated in terms of the one for the (m – 1)-th derivative. The rounding errors in an important case of Chebyshev polynomial are discussed in some detail.This revised version was published online in October 2005 with corrections to the Cover Date.