Abstract. This paper discusses a method based on Laguerre polynomials combined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponential of a matrix by a vector. The method implicitly uses an expansion of the exponential function in a series of orthogonal Laguerre polynomials, much like existing methods based on Chebyshev polynomials do. Owing to the fact that...

Claim of Johnson-Lindenstrauss Theorem: The Euclidean metric on any finite set X (a bunch of high dimensional points) can be embedded with distortion D = 1 + ε in R for k = O(ε−2 log n). If we lose ε (ε = 0), it becomes almost impossible to do better than that in R. Nevertheless, it is not hard to construct a counter example to this: a simplex of n+ 1 points. The Johnson-Lindenstrauss theorem g...

The following estimate for the Rayleigh–Ritz method is proved: |λ̃−λ||(ũ,u)| ≤ ‖Aũ− λ̃ũ‖sin∠{u;Ũ}, ‖u‖= 1. Here A is a bounded self-adjoint operator in a real Hilbert/euclidian space, {λ,u} one of its eigenpairs, Ũ a trial subspace for the Rayleigh–Ritz method, and {λ̃, ũ} a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh–Ritz method, in part...

We study quasi–Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of ε depends on ε−1 and the dimension s. Strong tractability means that it does not depend on s and is bounded by a polynomial in ε−1. The least po...

We give a nonstandard variant of Jordan’s proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is selfcontained, except for the Jordan theorem for polygons taken for granted.

and Applied Analysis 3 where A is an n × n constant matrix with n different eigenvalues λ1, λ2, . . . , λn and Q t is analytic quasi-periodic with respect to t with frequencies ω ω1, ω2, . . . , ωl . Here ε is a small perturbation parameter. Suppose that the following nonresonance conditions hold: ∣∣〈k,ω〉 √ −1 λi − λj ∣∣ ≥ α |k| , 1.8 for all k ∈ Z \ {0}, where α > 0 is a small constant and τ >...

We study interpolation polynomials based on the points in [−1, 1]× [−1, 1] that are common zeros of quasi-orthogonal Chebyshev polynomials and nodes of near minimal degree cubature formula. With the help of the cubature formula we establish the mean convergence of the interpolation polynomials. 1991 Mathematics Subject Classification: Primary 41A05, 33C50.

In this paper a new technique is proposed for the clustering and classification of spatio-temporal object trajectories extracted from video motion clips. The trajectories are represented as motion time series and modelled using Chebyshev polynomial approximations. Trajectory clustering is then performed to discover patterns of similar object motion. The coefficients of the basis functions are u...

Chebyshev functions, which are equiripple in a certain domain, are used to generate equiripple halfband lowpass frequency responses. Inverse Fourier transformation is then used to obtain explicit formulas for the corresponding impulse responses. The halfband lowpass FIR digital filters designed in this way are quasi-equiripple, having performances very close to those of true equiripple filters,...

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1. a simple and efficient algorithm that achieves an n−1/3-approximation; 2. NP-hardness of approximation to within (1− ε), for some small constant ε > 0; 3. SSE-hardness of approximatio...