顺序统计量的傅里叶变换及其在Lorentz空间中的应用
The Fourier transform of order statistics with applications to Lorentz spaces
摘要：
本文给出了$\Bbb R^n$中顺序统计量的Fourier变换公式，证明了在$\Bbb R^n$中的坐标平面外，所有这些Fourier变换都等于一个常数倍。对于$a_1\geq ... \geq a_n\ge0$和$q>0,$，用$\ell_{w,q}^n$表示具有$\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$范数的$n$维洛伦兹空间，其中，$(x_1^{*},...,x_n^{*})$为非利用上述公式和Banach空间等距可嵌入性的傅里叶变换准则，空间转化为$L_q$ \cite{10}，证明了对于$n\geq 3$和$q\leq 1,$，空间$\ell_{w,q}^n$与$L_q$的子空间等距当且仅当$a_1,...,a_n$形成等差数列。对于$q>1,$所有的数字$a_i$必须是相等的，使$\ell_{w,q}^n = \ell_q^n.$因此，洛伦兹函数空间$L_{w,q}(0,1)$是等距的一个子空间$L_q$当且仅当{ \it either} $0<q<\infty$和权重$w$是一个常数函数（使$L_{w,q}= L_q$），{ \it either} $q\le 1$和$w(t)$是一个递减线性函数。最后，我们把我们的结果与正定函数理论联系起来。
英文摘要：
We present a formula for the Fourier transforms of order statistics in $\Bbb R^n$ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in $\Bbb R^n.$ For $a_1\geq ... \geq a_n\ge0$ and $q>0,$ denote by $\ell_{w,q}^n$ the $n$-dimensional Lorentz space with the norm $\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into $L_q$ \cite{10} to prove that, for $n\geq 3$ and $q\leq 1,$ the space $\ell_{w,q}^n$ is isometric to a subspace of $L_q$ if and only if the numbers $a_1,...,a_n$ form an arithmetic progression. For $q>1,$ all the numbers $a_i$ must be equal so that $\ell_{w,q}^n = \ell_q^n.$ Consequently, the Lorentz function space $L_{w,q}(0,1)$ is isometric to a subspace of $L_q$ if and only if {\it either} $0<q<\infty$ and the weight $w$ is a constant function (so that $L_{w,q}= L_q$), {\it or} $q\le 1$ and $w(t)$ is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions.
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