اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLocal asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron
152
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a semi-rational polyhedron and a real sampling parameter t, if we allow for step-polynomial coefficients, instead of just constant ones.
قيم البحث
اقرأ أيضاً
We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with c
onstant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function D(N(F)).f, where we denote by N(F) the normal cone to P along F.
In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.
In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums md0 R_nf(x)=frac{1}{n}sum_{k=0}^{n-1}fbigg(x+frac{k}{n}bigg),quad xin ZT. emd We establish a relevant connection between Riemann and ordinary maxim
al functions, which allows to use techniques and results of the theory of differentiations of integrals in $ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $fin L^p$ with $p>1$.
We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by [S_ u(a)=sum_{ngeq 1} (frac{1}{2} an^2)^{- u} K_ u(an^2)qquad (|arg,a|<pi/2)] as the parameter $|a|to 0$. It is shown that the posit
ive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $ato 0$ through positive values. Numerical results are presented to support the asymptotic theory.
We study the finite field extension estimates for Hamming varieties $H_j, jin mathbb F_q^*,$ defined by $H_j={xin mathbb F_q^d: prod_{k=1}^d x_k=j},$ where $mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $mathbb F_q$ with
$q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2to L^r$ extension estimate for $H_j$ holds.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
