The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement

A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduced to a set of algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use.     

New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems

Let T:D⊂X→X$T:D\subset X\to X$ be an iteration function in a complete metric space X$X$. In this paper we present some new general complete convergence theorems for the Picard iteration _xn+1=T_xn$x_{n+1}=T{x}_n$ with order of convergence at least r≥1$r\ge 1$. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of T$T$ and a convergence function   of T$T$. We study the convergence of the Picard iteration associated to T$T$ with respect to a function of initial conditions E:D→X$E:D\to X$. The initial conditions in our convergence results utilize only information at the starting point _x0$x_0$. More precisely, the initial conditions are given in the form E(_x0)∈J$E\left(x_0\right)\in J$, where J$J$ is an interval on _R+${\mathbb{R}}_{+}$ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω$\omega$-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal α$\alpha$-theorem of Smale for analytic functions.     

Approximation of Sobolev classes by polynomials and ridge functions

Let be the usual Sobolev class of functions on the unit ball in , and be the subclass of all radial functions in . We show that for the classes and , the orders of best approximation by polynomials in coincide. We also obtain exact orders of best approximation in of the classes by ridge functions and, as an immediate consequence, we obtain the same orders in for the usual Sobolev classes .     

Density results with linear combinations of translates of fundamental solutions

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of . The domains under consideration may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces and the spaces of uniformly Hölder continuous functions , and we establish density and non-density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series.     

Endpoint estimates for some maximal operators associated to the circular conical surface

For and m>0 we consider the maximal operator given by

     

On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong-Loewy conjecture

Let C={1,2,…,m} and f be a multiplicative function such that (f∗μ)(k)>0 for every positive integer k and the Euler product converges. Let (C_f)=(f(i,j)) be the m×m matrix defined on the set C having f evaluated at the greatest common divisor (i,j) of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of (C_f)^-1, the inverse of (C_f), in terms of ζ_f. Specializing these bounds for the arithmetical functions f=N^ε,J_ε and σ_ε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (C_N^ε),(C_Jε) and (C_σε) depending on ε when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy [S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551-569].     

On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.