اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn estimate of the root mean square error incurred when approximating an $f in L^2({mathbb{R}})$ by a partial sum of its Hermite series
210
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $f$ be a band-limited function in $L^2({mathbb{R}})$. Fix $T >0$ and suppose $f^{prime}$ exists and is integrable on $[-T, T]$. This paper gives a concrete estimate of the error incurred when approximating $f$ in the root mean square by a partial sum of its Hermite series. Specifically, we show, for $K=2n, quad n in Z_+,$ $$ left[frac{1}{2T}int_{-T}^T[f(t)-(S_Kf)(t)]^2dtright]^{1/2}leq left(1+frac 1Kright)left(left[ frac{1}{2T}int_{|t|> T}f(t)^2dtright]^{1/2} +left[frac{1}{2T} int_{|omega|>N}|hat f(omega)|^2domegaright]^{1/2} right) +frac{1}{K}left[frac{1}{2T}int_{|t|leq T}f_N(t)^2dtright]^{1/2} +frac{1}{pi}left(1+frac{1}{2K}right)S_a(K,T), $$ in which $S_Kf$ is the $K$-th partial sum of the Hermite series of $f, hat f $ is the Fourier transform of $f$, $displaystyle{N=frac{sqrt{2K+1}+% sqrt{2K+3}}{2}}$ and $f_N=(hat f chi_{(-N,N)})^vee(t)=frac{1}{pi}int_{-infty}^{infty}frac{sin (N(t-s))}{t-s}f(s)ds$. An explicit upper bound is obtained for $S_{a}(K,T)$.
قيم البحث
اقرأ أيضاً
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good an
d bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trili
near oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
Using Guths polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrodinger equation in $mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $lim_{t
to 0} e^{itDelta}f(x)=f(x)$ almost everywhere for all $f in H^s (mathbb{R}^2)$ provided that $s>3/8$.
Recent years have witnessed a controversy over Heisenbergs famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss
two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for {em state-dependent} errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for {em state-independent} errors have been proven.
In this paper, we prove two improv
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
