﻿ 多元Whittaker-Shannon采样展开的截断误差

# 多元Whittaker-Shannon采样展开的截断误差Truncation Errors for Multi-Dimensional Whittaker-Shannon Sampling Expansion

Abstract: In this paper, we study a general model that uses linear functionals to cover several errors in one formula, consider sampling series with measured sampled values for band limited signals without decay assumption, and obtain the optimal bounds of truncation errors for band limited signal functions from Paley-Wiener spaceB2V(ℝd).

1. 引言

${L}_{p}\left({ℝ}^{d}\right)$$1\le p\le \infty$ ，表示赋予以下范数的全体在 ${ℝ}^{d}$ 上具有p次勒贝格可积函数所组成的空间。其范数可表示为

${‖f‖}_{{L}_{p}}:=\left\{\begin{array}{l}{\left(\underset{{R}^{d}}{\int }|f{\left(t\right)}^{p}|\text{d}t\right)}^{1/p},\text{\hspace{0.17em}}\text{ }1\le p<\infty ,\\ \underset{t\in {R}^{d}}{\text{esssup}}|f\left(t\right)|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}p=\infty .\end{array}$

${Z}_{d}=\left\{1,2,3,\cdots ,d\right\}$ 。如果对于任意的正坐标向量 $v:=\left({\upsilon }_{j}:j\in {Z}_{d}\right)$ ，且对 $\forall \epsilon >0$ 都存在一个正整数c，使得对所有的复向量 $z:=\left({z}_{j}:j\in {Z}_{d}\right)\in {ℂ}^{d}$ ，有

$|h\left(z\right)|\le c\mathrm{exp}\left(\underset{j\in {Z}_{d}}{\sum }\left({\upsilon }_{j}+\epsilon \right)|{z}_{j}|\right).$

${B}_{v}^{p}\left({ℝ}^{d}\right):={B}_{v}\left({ℝ}^{d}\right)\cap {L}_{p}\left({ℝ}^{d}\right).$

$v=\upsilon e,\upsilon \in {R}_{+}$ ，我们用 ${I}_{\upsilon }^{d}$ 来表示。根据Schwartz定理，我们有

${B}_{v}^{p}\left({R}^{d}\right)=\left\{f\in {L}_{p}\left({R}^{d}\right):\mathrm{sup}p\stackrel{^}{f}\subseteq {I}_{v}^{d}\right\}$

$f\left(t\right)=\left({S}_{v}f\right)\left(t\right):=\underset{k\in {ℤ}^{d}}{\sum }f\left(\frac{k}{v}\right)\mathrm{sin}c\left(v\cdot t-k\right),$ (1)

$|f\left(t\right)-\underset{|{k}_{d}|\le {N}_{d}}{\sum }\cdots \underset{|{k}_{1}|\le {N}_{1}}{\sum }f\left(\frac{k}{v}\right)\mathrm{sin}c\left(v\cdot t-k\right)|$

${\Omega }_{v}\left(f,\lambda \right):=\mathrm{sup}|{\lambda }_{k}f\left(\cdot +k/v\right)-f\left(k/v\right)|,\Omega >0$

$\left({S}_{v}^{\lambda }f\right)\left(t\right):=\sum {\lambda }_{k}f\left(\cdot +k/v\right)\mathrm{sin}c\left(v\cdot t-k\right).$

$\left({S}_{v,N}^{\lambda }f\right)\left(t\right):=\underset{\text{k-v}\cdot \text{t}\in {I}_{N}^{d}}{\sum }{\lambda }_{k}f\left(\cdot +k/v\right)\mathrm{sin}c\left(v\cdot t-k\right),$

2. 有限带函数的截断误差

${\left(\frac{1}{{\prod }_{j=1}^{d}{\upsilon }_{j}}\underset{k\in {ℤ}_{d}}{\sum }{|f\left(\frac{k}{v}\right)|}^{p}\right)}^{1/p}\le C{‖f‖}_{{L}_{p}}$ . (2)

$\left({E}_{\upsilon e,Ne}^{\lambda }f\right)\left(t\right)\le C{\upsilon }^{-r+d/p}{\mathrm{ln}}^{d}\upsilon ,$

${\underset{k\in {ℤ}^{d}}{\sum }|\mathrm{sin}c\left(t-k\right)|}^{q}\le {\left(\frac{q}{q-1}\right)}^{d}.$ (3)

$\left({E}_{\upsilon e,Ne}^{\lambda }f\right)\left(t\right)\le {I}_{1}+{I}_{2}$

${I}_{1}={\left(\underset{v\cdot t-k\notin {I}_{Ne}^{d}}{\sum }{|f\left(\frac{k}{\upsilon }\right)|}^{p}\right)}^{1/p}{\left({\underset{v\cdot t-k\notin {I}_{Ne}^{d}}{\sum }|\mathrm{sin}c\left(\upsilon \cdot t-k\right)|}^{q}\right)}^{1/q},$ (4)

${I}_{2}={\left(\underset{v\cdot t-k\in {I}_{Ne}^{d}}{\sum }{|f\left(\frac{k}{\upsilon }\right)-{\lambda }_{k}f\left(\cdot +k/v\right)|}^{p}\right)}^{1/p}{\left(\underset{v\cdot t-k\in {I}_{Ne}^{d}}{\sum }{|\mathrm{sin}c\left(\upsilon \cdot t-k\right)|}^{q}\right)}^{1/q}.$ (5)

$\frac{1}{p}+\frac{1}{q}=1,p\ge 1$

$h\left(t\right)={\left(\underset{v\cdot t-k\notin {I}_{Ne}^{d}}{\sum }{|\mathrm{sin}c\left(\upsilon \cdot t-k\right)|}^{q}\right)}^{1/q},$

$\left\{k:k\notin {I}_{Ne}^{d}\right\}\subset \underset{j=1}{\overset{d}{\cup }}\left\{k:{k}_{j}\notin \left[-N,N\right]\right\}$

$\underset{v\cdot t-k\notin {I}_{Ne}^{d}}{\sum }{|\mathrm{sin}c\left(\upsilon \cdot t-k\right)|}^{q}\le \underset{j=1}{\overset{d}{\sum }}\underset{{k}_{j}\notin \left(-N,N\right]}{\sum }{|\mathrm{sin}c\left(\upsilon {t}_{i}-{k}_{j}\right)|}^{q}\underset{i\in {Z}_{d}\j}{\prod }\underset{{k}_{i}\in ℤ}{\sum }{|\mathrm{sin}c\left(\upsilon {t}_{i}-{k}_{i}\right)|}^{q}.$ (6)

$l\left(t\right)={\left(\underset{k\notin \left(-N,N\right]}{\sum }{|\mathrm{sin}c\left(\upsilon t-k\right)|}^{q}\right)}^{1/q}.$

$l\left(t\right)\le {\left(C\underset{k\notin \left(-N,N\right]}{\sum }\frac{1}{{|k|}^{q}}\right)}^{1/q}\le {\left(C\cdot {\int }_{N}^{\infty }{t}^{-q}\text{d}t\right)}^{1/q}\le C\cdot {N}^{-1/P}.$

${I}_{1}\le C\cdot {\left(\frac{{\upsilon }^{d}}{N}\right)}^{1/p}{‖f‖}_{{L}_{p}}.$ (7)

$\underset{v\cdot t-k\in {I}_{Ne}^{d}}{\sum }|\mathrm{sin}c\left(\upsilon \cdot t-k\right)|\le C\cdot {\mathrm{ln}}^{d}N.$

${I}_{2}\le C\cdot {\left(2N+1\right)}^{d/p}\Omega \cdot {\mathrm{ln}}^{d}N.$ (8)

$\left({E}_{\upsilon e,Ne}^{\lambda }f\right)\left(t\right)\le C\left({\left({\upsilon }^{d}/N\right)}^{1/P}{‖f‖}_{{L}_{p}}+{N}^{d/p}\Omega \cdot {\mathrm{ln}}^{d}N\right).$ (9)

$N=\left[{\upsilon }^{rp}\right]$ ，则

$\left({E}_{ve,Ne}^{\lambda }f\right)\left(t\right)\le C{\upsilon }^{-r+d/p}{\mathrm{ln}}^{d}\upsilon .$

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