Uncertainty Inequalities for Hankel Transforms
Uncertainty Inequalities for Hankel Transforms
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DOI：
10.1137/0502059
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发表时间：
1971-11
影响因子：
2
通讯作者：
P. C. Bowie
中科院分区：
文献类型：
--
作者：
P. C. Bowie
本文得到了Hankel变换的一个不确定不等式，设$\nu>；0$固定。我们设置[d\u_\nu(X)=c_\nu^{-1}x^{2v}dx，\quad c_\nu=2^{\nu-1}/2}}\Gamma(\nu+\frc{1}{2})，\]和\[{\bf J}_\nu(X)=c_\nu x^{-\nu+{1/2}}J_{\nu-{1/2}}(X)，其中$J_{\nu-{1/2}}(X)$是第一类贝塞尔函数$\nu-\FRAC{1}{2}$。我们定义\[f^\楔形(t；N)=int_0^\inty{f(X)J_n u(Xt)d_u_n(X)}.关于$d_u_n$的概率频率函数定义为$L_n^1(0，\inty)$中范数为1的非负函数，概率频率函数$F(X)$的广义方差定义为[V_nu[F]=int_0^\inty{x^2 F(X)d_u(X)}.设$f(X)$属于范数为1的$L_nu^2(0，\inty)$.根据Parseval等式，可将$|{f(X)}|^2$和$|{f^\wedge(x；v)}|^2$视为概率频率函数。不确定性不等式[V_\nu\Left[{|{f(X)}|^2}]V_\nu[{|{f^\wedge(x；...
In this paper an uncertainty inequality for Hankel transforms is obtained.Let $\nu > 0$ be fixed. We set \[ d\mu _\nu (x) = c_\nu ^{ - 1} x^{2v} dx,\quad c_\nu = 2^{{{\nu - 1} / 2}} \Gamma (\nu + \frac{1}{2}),\] and \[ {\bf J}_\nu (x) = c_\nu x^{ - \nu + {1 / 2}} J_{\nu - {1 / 2}} (x),\] where $J_{\nu - {1 / 2}} (x)$ is a Bessel function of the first kind of order $\nu - \frac{1}{2}$. We define \[ f^ \wedge (t;\nu ) = \int_0^\infty {f(x)J_\nu (xt)d\mu _\nu (x)} .\]A probability frequency function with respect to $d\mu _\nu $, is defined as a nonnegative function in $L_\nu ^1 (0,\infty )$ with norm one, and the generalized variance of a probability frequency function $F(x)$ is defined by \[ V_\nu [F] = \int_0^\infty {x^2 F(x)d\mu _\nu (x)} .\] Let $f(x)$ belong to $L_\nu ^2 (0,\infty )$ with norm one. By Parseval’s equality $| {f(x)} |^2 $ and $| {f^\wedge (x;v)} |^2 $ can be considered as probability frequency functions. The uncertainty inequality \[ V_\nu \left[ {| {f(x)} |^2 } ]V_\nu [ {| {f^ \wedge (x;...