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We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave in terms of spherical harmonics . We consider the truncated series where the summation is performed over the ’s satisfying . We prove that if is large enough, the truncated series gives rise to an error lower than as soon as satisfies where is the Lambert function and are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results...
We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, , which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices . We prove that if is large enough, the truncated series gives rise to an error lower than as soon as satisfies where is the Lambert function, depends only on and are...
We perform a complete study
of the truncation error of the Gegenbauer series.
This series yields an expansion of the Green kernel of the
Helmholtz equation,
,
which is the core of the Fast Multipole Method for the integral equations.
We consider the truncated series where the summation is
performed over the indices .
We prove that if is large enough,
the truncated series gives rise to an error lower than ϵ
as soon as L satisfies
where W is the Lambert function,
depends only on...
We perform a complete study
of the truncation error of the Jacobi-Anger series.
This series expands every
plane wave in terms of
spherical harmonics
.
We consider the truncated series where the summation is
performed over the 's satisfying .
We prove that if is large enough,
the truncated series gives rise to an error lower than ϵ
as soon as L satisfies
where W is the Lambert function and
are pure positive constants.
Numerical experiments show that this
asymptotic is...
Let || · || be the uniform norm in the unit disk. We study the quantities Mn (α) := inf (||zP(z) + α|| - α) where the infimum is taken over all polynomials P of degree n - 1 with ||P(z)|| = 1 and α > 0. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that infα>0Mn (α) = 1/n. We find the exact values of Mn (α) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
The aim of this article is to give new refinements and sharpenings of Shafer's inequality involving the arctangent function. These are obtained by means of a change of variables, which makes the computations much easier than the classical approach.
Let {bH(t), t∈ℝ} be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type X(t)=c+∫0tσ(X(u)) dbH(u)+∫0tμ(X(u)) du. In different particular models where σ(x)=σ or σ(x)=σ
x and μ(x)=μ or μ(x)=μ
x, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(⋅)...
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