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We consider the numerical computation of Hadamard Finite Part (HFP) integrals H σ (t ; u) = ⨎ 0 T | sin π (x − t) T | σ u (x) d x , 0 < t < T , σ < − 1 , σ ⁄ ∈ Z , u (x) being sufficiently differentiable and T -periodic on R. Thus σ = − (m + δ) , m ∈ { 1 , 2 , 3 , ... } , 0 < δ < 1. For each such σ , we regularize H σ (t ; u) , and show that H σ (t ; u) = H σ + 2 r (t ; U σ) , r = ⌊ (m + 1) / 2 ⌋ , where U σ (x) = ∑ k = 0 r a k u (2 k) (x) for some constants a k , H σ + 2 r (t ; U σ) being a regular integral. We then propose to approximate H σ (t ; u) by the quadrature formula Q σ , n (t ; u) ≡ H σ (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q σ , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q σ , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals H σ (t ; u) with σ < − 1 and σ ⁄ ∈ Z apply, with no changes, to the regular singular integrals H σ (t ; u) with σ > − 1 and σ ⁄ ∈ Z. We illustrate the effectiveness of Q σ , n (t ; u) with numerical examples both for σ < − 1 and σ > − 1. Finally, we show that the HFP or regular integral ⨎ 0 T f (x) d x of any T -periodic integrand f (x) that has algebraic singularities of the form | x − t + k T | σ , 0 < t < T , k = 0 , ± 1 , ± 2 , ... , with σ ⁄ ∈ Z , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed as H σ (t ; u) , where u (x) is a T -periodic and sufficiently differentiable function of x on R that can be computed from f (x). Therefore, ⨎ 0 T f (x) d x can be computed efficiently using our new numerical quadrature formulas Q σ , n (t ; u) on the individual H σ (t ; u). Again, only 2 n function evaluations, namely, u (x n , k) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a -periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in and - norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. [Copyright &y& Elsevier]

Abstract: We discuss the effect of a particular sequence acceleration method, the process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points. [Copyright &y& Elsevier]