数学杂志  2021, Vol. 41 Issue (6): 489-494   PDF    
扩展功能
加入收藏夹
复制引文信息
加入引用管理器
Email Alert
RSS
本文作者相关文章
YU An-kang
LI Bao-de
APPROXIMATIONS OF THE IDENTITY ADAPTED TO CONTINUOUS ELLIPSOID COVER
YU An-kang, LI Bao-de    
School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Abstract: In this paper we develop some approximation of the identity results adapted to continuous multi-level ellipsoid cover. By using real-variable methods of harmonic analysis, we obtain two approximations of the identity results uniformly in some compact subset of $ \mathbb{R}^n$ and in $ L^1(\mathbb{R}^n)$ norm, respectively. These results generalize the corresponding classical and anisotropic approximation of the identity results.
Keywords: approximation of the identity     ellipsoid cover     anisotropy    
相适应于连续椭球覆盖的恒等逼近
余安康, 李宝德    
新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046
摘要:本文研究了相适应于多尺度连续椭球覆盖的恒等逼近问题. 通过使用调和分析中的实变方法,得到了如下两个恒等逼近结果:在欧氏空间$ \mathbb{R}^n$上紧子集的一致恒等逼近和$ L^1(\mathbb{R}^n)$范数下的恒等逼近. 该结果推广了经典情况下和各向异性情形下相应的恒等逼近结果.
关键词恒等逼近    椭球覆盖    各向异性    
1 Introduction

As we all know, approximation of identity plays an important role in analysis, see [1-3]. There are numerous approximations of identity results associated with the Euclidian balls in $ \mathbb{R}^n $. For example, let $ \varphi $ be an integrable function on $ \mathbb{R}^n $ such that $ \int_{\mathbb{R}^n}\varphi(x)\, dx = 1 $, and for $ t>0 $ define $ \varphi_t(x) = t^{-n}\varphi(t^{-1}x) $. Then, if $ f\in L^1(\mathbb{R}^n) $, $ \varphi_t\ast f\rightarrow f\ (t\rightarrow 0) $ in $ L^1(\mathbb{R}^n) $.

In 2010, the continuous multi-level ellipsoid cover $ \Theta $ introduced by Dahmen, Dekel and Petrushev [4] consist of ellipsoids $ \theta_{x, t} = M_{x, t}(\mathbb{B}^n)+x $, where $ M_{x, t} $ is an invertible matrix and $ \mathbb{B}^n $ is the unit ball in $ \mathbb{R}^n $ (see Definition 2.1). The flexible framework of continuous ellipsoid cover $ \Theta $ introduced in this paper may have the ability to solve the following problems. For example, the formation of shocks results in jump discontinuities of solutions of hyperbolic conservation laws across lower dimensional manifolds. The case such jumps cause a serious obstruction to appropriate regularity theorems, since the available regularity scales are either inherently isotropic or coordinate biased or are subject to an uncontrollable restricted regularity range. For more development of continuous ellipsoid cover, see [5-7].

Inspired by the above work, for any $ \theta_{x, t} = M_{x, t}(\mathbb{R}^n)+x\in \Theta $, let $ \varphi $ be an integrable function on $ \mathbb{R}^n $ such that $ \int_{\mathbb{R}^n}\varphi\, dx = 1 $, we can define

$ \varphi_{x, \, t}(y): = |{\text{det}}(M^{-1}_{x, \, t})|\varphi(M^{-1}_{x, \, t}y). $

And then a question arises: Is it possible to obtain some approximations of the identity results adapted to ellipsoid cover $ \Theta $ such as $ f\ast\varphi_{x, t}(x)\rightarrow f(x) $ ($ t\rightarrow \infty $) in various senses? This article gives some affirmative answers for the question. It is worth pointing out that the approximation of the identity in this paper is done in $ C_c(\mathbb{R}^n) $, which is a dense subset of $ L^1(\mathbb{R}^n) $, and the approximation of the identity in $ L^1(\mathbb{R}^n) $ is difficult for us, which is still open at the moment.

The organization of this article is as follows. In Section 2, we first present some notation and notions used in this article including continuous ellipsoid cover $ \Theta $ and describe our main theorem. In Section 3, we show the proof details of the main theorem.

2 Preliminaries and Main Results

In this section we recall the properties of ellipsoid cover which was originally introduced by Dahmen, Dekel, and Petruschev [4]. An ellipsoid $ \xi $ in $ \mathbb{R}^n $ is an image of the Euclidean unit ball $ \mathbb{B}^n: = \{x\in\mathbb{R}^n: |x|<1\} $ under an affine transform, i.e.,

$ \xi: = M_\xi(\mathbb{B}^n)+c_\xi, $

where $ M_\xi $ is an invertible matrix and $ c_\xi $ is the center.

Definition 2.1  We say that

$ \Theta: = \{ \theta_{x, \, t}: x\in\mathbb{R}^n, t\in\mathbb{R}\} $

is a continuous ellipsoid cover of $ \mathbb{R}^n $, if there exist constants $ {\bf{p}}(\Theta): = \{a_1, \ldots, a_6\} $ such that:

${\rm{ (i)}} $ For every $ x\in \mathbb{R}^n $ and $ t\in \mathbb{R} $, there exists an ellipsoid $ \theta_{x, \, t}: = M_{x, \, t}(\mathbb{B}^n)+x $, where $ M_{x, \, t} $ is a invertible matrix and $ x $ is the center, satisfying

$ \begin{eqnarray} a_12^{-t}\leq|\theta_{x, \, t}|\leq a_2 2^{-t}. \end{eqnarray} $ (2.1)

$ {\rm{(ii)}} $ Intersecting ellipsoids from $ \Theta $ satisfy "shape condition", i.e., for any $ x, \, y\in \mathbb{R}^n $, $ t\in \mathbb{R} $ and $ s\ge0 $, if $ \theta_{x, \, t}\cap \theta_{y, \, t+s}\ne\emptyset $, then

$ \begin{eqnarray} a_3 2^{-a_4 s}\le 1 / \| (M_{y, \, t+s})^{-1} M_{x, \, t}\| \le \|(M_{x, \, t})^{-1} M_{y, \, t+s}\| \le a_5 2^{-a_6 s}. \end{eqnarray} $ (2.2)

Here, $ \|\cdot \| $ is the matrix norm given by $ \|M\|: = \max_{|x| = 1}|Mx| $ for invertible matrix $ M $.

Proposition 2.2  For any $ x\in\mathbb{R}^n $ and $ \{\theta_{x, t}\}_{t\in \mathbb{R}}\subset \Theta, $ it holds true that

$ \begin{align} \theta_{x, t} = M_{x, \, t}(\mathbb{B}^n)+x\to \mathbb{R}^n\, \, \, \, \mathrm{as}\, \, \, \, t\to-\infty. \end{align} $ (2.3)

Proof  For any $ y\in\mathbb{B}^n $ and $ t<0 $, by (2.2), we obtain

$ \begin{eqnarray*} |M^{-1}_{x, \, t}y|\le \|M^{-1}_{x, \, t}M_{x, \, 0}\| \|M^{-1}_{x, \, 0}\| |y| \le a_52^{a_6t}\|M^{-1}_{x, \, 0}\| |y|< a_52^{a_6t}\|M^{-1}_{x, \, 0}\|, \end{eqnarray*} $

which implies that

$ \begin{eqnarray*} a_5^{-1}\|M^{-1}_{x, \, 0}\|^{-1}2^{-a_6t}\mathbb{B}^n\subset M_{x, \, t}(\mathbb{B}^n) \end{eqnarray*} $

and hence $ \theta_{x, t} = M_{x, \, t}(\mathbb{B}^n)+x\to \mathbb{R}^n $ as $ t\to-\infty $.

Definition 2.3  For each $ x\in\mathbb{R}^n $, $ t\in\mathbb{R} $ and $ \theta_{x, \, t} = M_{x, \, t}(\mathbb{B}^n)+x\in\Theta $, denote

$ \varphi_{x, \, t}(y): = |{\text{ det}}(M^{-1}_{x, \, t})|\varphi(M^{-1}_{x, \, t}y) $

and, for each measurable function $ f $,

$ f\ast\varphi_{x, t}(x): = \int_{\mathbb{R}^n}f(y)\varphi_{x, t}(x-y)dy. $

Theorem 2.4

$ \rm{(i)} $ Let $ \int_{\mathbb{R}^n}\varphi(x)\, dx = 1 $ and $ f\in C_c(\mathbb{R}^n) $, which collects all continuous functions of compact support. Then

$ \lim\limits_{t\rightarrow +\infty}\sup\limits_{x\in \text{supp} f}|f\ast\varphi_{x, t}(x)-f(x)| = 0. $

$ \rm{(ii)} $ Let $ f\in C_c(\mathbb{R}^n) $, $ \int_{\mathbb{R}^n}\varphi(x)\, dx = 1 $ and $ \text{supp} \varphi\subset\mathbb{B}^n $. Moreover, if the cover $ \Theta $ is zero-level-bounded, i.e., there exists a positive constant $ C $ such that, for any $ x\in \mathbb{R}^n $ and matrix $ M_{x, 0} $ of ellipsoid $ \theta_{x, 0} = M_{x, 0}(\mathbb{B}^n)+x $,

$ \begin{equation} \|M_{x, 0}\|\leq C. \end{equation} $ (2.4)

Then

$ \lim\limits_{t\rightarrow +\infty}\int_{\mathbb{R}^n}|f\ast\varphi_{x, t}(x)-f(x)|\, dx = 0. $

Remark 2.5  The assumption (2.4) is a mild condition. When the ellipsoid cover $ \Theta $ is reduced to the following typical ellipsoid covers, the assumption (2.4) holds true automatically.

$ {\rm{(i)}} $ The regular cover of $ \mathbb{R}^{n} $ consisting of all balls in $ \mathbb{R}^{n} $:

$ \begin{equation*} \Theta: = \{\theta(x, \, t) = 2^{-\frac{t}{n}}\text{I}_{n}(\mathbb{B}^{n})+x: t\in \mathbb{R}, \, x\in\mathbb{R}^{n}\}. \end{equation*} $

Obviously, $ \|M_{x, 0}\| = 1 $.

$ {\rm{(ii)}} $ The one-parameter family of diagonal dilation matrices

$ \begin{equation*} D_{t}: = \text{diag}(2^{-tb_{1}}, 2^{-tb_{2}}, \cdot\cdot\cdot, 2^{-tb_{n}}), \ \ b_{j}>0, \, j = 1, \cdot\cdot\cdot, n, \, \sum\limits_{j = 1}^{n}b_{j} = 1, \end{equation*} $

induces a continuous ellipsoid cover of $ \mathbb{R}^{n} $ with $ M_{x, \, t} = D_{t} $. Obviously, $ \|M_{x, 0}\| = 1 $.

$ {\rm{(iii)}} $ For a one parameter continuous subgroup of $ GL(\mathbb{R}^n, n) $ of the form $ \{A_t:0<t<\infty\} $ satisfying $ A_t A_s = A_{ts} $ and, there exist constants $ 1\leq \alpha\leq\beta<\infty $ such that, $ t^{\alpha}|x|<|A_tx|<t^{\beta}|x| $ for all $ x\in\mathbb{R}^n $ and $ t\geq 1 $. The infinitesimal generator $ P $ of $ A_t: = \exp(P\ln t) $ satisfies $ (Px, x)\geq (x, x) $ and $ \det A_{t} = t^{a} $, where $ (\cdot, \cdot) $ is the standard scalar product in $ \mathbb{R}^n $ and $ a $ is trace of $ P $. Then we can define a continuous ellipsoid cover $ \Theta $ of Calderón and Torchinsky [8], that is,

$ \Theta: = \{x+A_t(\mathbb{B}^n):x\in\mathbb{R}^n, t\in (0, \infty)\}: = \{x+A_{2^{-t/a}}(\mathbb{B}^n): x\in\mathbb{R}^n, t\in\mathbb{R}\}. $

Obviously, $ \|M_{x, 0}\| = \|A_1\| = \|\exp (P\ln1)\| = 1 $.

$ {\rm{(iv)}} $ Consider a $ n\times n $ real matrix $ A $ with eigenvalues $ \lambda $ satisfying $ |\lambda| > 1 $. By [9, Lemma 2.2], there exists an ellipsoid $ \Delta: = \{x\in\mathbb{R}^n:\, |Px|<1\} $, where $ P $ is some invertible $ n\times n $ matrix, such that $ B_k\subset B_{k+1} $, where $ B_k: = A^k\Delta $ for $ k\in \mathbb{Z} $. Then we can define a semi-continuous ellipsoid cover in the sense of [4, Definition 2.5] by

$ \begin{equation*} \Theta: = \{x+B_k:\, x\in\mathbb{R}^n, k\in\mathbb{Z}\} = \{x+A^{k}P^{-1}(\mathbb{B}^n):\, x\in\mathbb{R}^n, k\in\mathbb{Z}\}. \end{equation*} $

Obviously, $ \|M_{x, 0}\| = \|P^{-1}\| $.

$ {\rm{(v)}} $ Lighted by [4, Theorem 7.3], we give a concrete example for the ellipsoid cover $ \Theta $:

$ \Theta_0: = \{\theta_{x, \, t} = M_{x, \, t}(\mathbb{B}^{n})+x:\ x\in \mathbb{R}^2, \, t\in\mathbb{R}\} $

with

$ \theta_{x, \, t}: = \{z = (z_1, \, z_2)\in \mathbb{R}^2:\ \frac{(z_1-x_1)^2}{{\sigma}^2_1} +\frac{(z_2-x_2)^2}{{\sigma}^2_2}\le 1\}, $

where

Here, $ M_{x, \, t} = \text{diag}(\sigma_1, \, \sigma_2) $. Obviously, $ \|M_{x, \, 0}\| = \|\text{I}_{n\times n}\| = 1 $,

3 Proof of Theorem 2.4

Proof  $ {\rm{(i)}}$ By $ f\in C_c({\mathbb{R}^n}), $ we know that there exists a positive constant $ N $ and $ M $ such that $ \text{supp} f\subset N \mathbb{B}^n $ and $ |f(x)|\leq M $ for any $ x\in \mathbb{R}^n $, and $ f $ is a uniformly continuous function on $ \mathbb{R}^n $. By this, we obtain that, for any $ \varepsilon>0 $, there exists $ \delta>0 $ such that, for all $ x\in \mathbb{R}^n, \ y\in \mathbb{R}^n $ with $ |y|<\delta, $

$ |f(x-y)-f(x)|<{\varepsilon}/({2\|\varphi\|_1}). $

From this, we deduce that, for any $ x\in \text{supp} f, $

$ \begin{align} |f\ast\varphi_{x, t}(x)-f(x)| & \leq\int_{|y|<\delta}|f(x-y)-f(x)||\varphi_{x, t}(y)|\, dy+ \int_{|y|\geq\delta}|f(x-y)-f(x)||\varphi_{x, t}(y)|\, dy\\ & <\frac{\varepsilon}{2}+2M\int_{|M_{x, t}y|\geq\delta}|\varphi(y)|\, dy. \end{align} $ (3.1)

From Proposition 2.2, it follows that there exists a positive constant $ t_N $ large enough such that $ N\mathbb{B}^n\subset \theta_{0, -t_N} $ and hence, for any $ x\in N\mathbb{B}^n $, $ x\in \theta_{0, -t_N} $. By this and (2.2), we obtain

$ \|M_{x, 0}\|\leq \|M_{0, -t_N}\|\|M_{0, -t_N}^{-1}M_{x, 0}\|\leq \|M_{0, -t_N}\|a_52^{-a_6t_N} = :C_N. $

From this and (2.2), for any $ x\in N\mathbb{B}^n $, $ y\in \mathbb{R}^n $ and $ t>0 $, we deduce that

$ |M_{x, t}y| = |M_{x, 0}M^{-1}_{x, 0}M_{x, t}y|\leq C_Na_{5}2^{-a_{6}t}|y|, $

which implies that

$ \{y:|M_{x, t}y|\geq \delta\}\subset \{y:|y|\geq (C_Na_5)^{-1} 2^{a_6t}\delta\}. $

By this and $ \varphi\in L^1(\mathbb{R}^n) $, we know that, for any $ \varepsilon>0 $, there exists a positive constant $ T $ large enough such that, for any $ t>T $,

$ \int_{|M_{x, t}y|\geq \delta}|\varphi(y)|dy\leq\int_{|y|\geq (C_Na_5)^{-1} 2^{a_6t}\delta}|\varphi(y)|dy<\frac{\varepsilon}{4M}, $

which, together with (3.1), implies that

$ \lim\limits_{t\rightarrow +\infty}\sup\limits_{x\in \text{supp} f}|f\ast\varphi_{x, t}(x)-f(x)| = 0. $

$ {\rm{(ii)}}$ By $ f\in C_c(\mathbb{R}^n) $, we can assume that there exists $ N>0 $ such that $ \text{supp} f \subset N\mathbb{B}^n $. Notice that

$ \begin{equation} \int_{\mathbb{R}^n}|f\ast\varphi_{x, t}(x)-f(x)|\, dx\leq\int_{\mathbb{B}^n}|\varphi(y)|\int_{\mathbb{R}^n} |f(x-M_{x, t}y)-f(x)|\, dx\, dy. \end{equation} $ (3.2)

Since $ \text{supp} f\subset N\mathbb{B}^n $, we only need to estimate the above integral under the following condition:

$ y\in \mathbb{B}^n\ \text{and}\ x\in \{x:x-M_{x, t}y\in N\mathbb{B}^n\}\cup N\mathbb{B}^n. $

For any $ t>0 $, by this, (2.2) and (2.4), we have

$ |x|\leq |M_{x, t}y|+N\leq \|M_{x, 0}\|a_52^{-a_6t}+N\leq Ca_5+N $

and hence $ x\in (Ca_5+N)\mathbb{B}^n $. From this, we further deduce that

$ \int_{\mathbb{R}^n}|f(x-M_{x, t}y)-f(x)|\, dx\leq \int_{(Ca_5+N)\mathbb{B}^n}|f(x-M_{x, t}y)-f(x)|\, dx. $

By this, (3.2), Fatou's lemma, the uniformly continuity of $ f $ on $ (Ca_5+N)\mathbb{B}^n $ and the fact that

$ |M_{x, t}y|\leq \|M_{x, 0}\|a_52^{-a_6t} \leq Ca_5 2^{-a_6t}\rightarrow 0\ \ \text{as}\ t\rightarrow +\infty, $

we have

$ \begin{align} {\limsup\limits_{t\rightarrow +\infty}}\int_{\mathbb{R}^n}|f\ast\varphi_{x, t}(x)-f(x)|\, dx & \leq {\limsup\limits_{t\rightarrow +\infty}}\int_{\mathbb{B}^n}|\varphi(y)|\int_{(Ca_5+N)\mathbb{B}^n} |f(x-M_{x, t}y)-f(x)|\, dx\, dy \\ & \leq \int_{\mathbb{B}^n}|\varphi(y)|\int_{(Ca_5+N)\mathbb{B}^n} {\limsup\limits_{t\rightarrow +\infty}}|f(x-M_{x, t}y)-f(x)|\, dx\, dy\\ & = 0. \end{align} $

This finishes the proof of Theorem 2.4(ⅱ).

References
[1]
Stein E M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[M]. Princeton: Princeton University Press, 1993.
[2]
Grafakos L. Classical fourier analysis, Second edition[M]. New York: Springer-Verlag, 2008.
[3]
Niu Tongtong, Wu Garidi. On approximation of non-negative continuous function by reciprocals of polynomial with positive coefficients in Orlicz spaces[J]. J. of Math, 2014, 34(1): 161-167.
[4]
Dahmen W, Dekel S, Petrushev P. Two-level-split decomposition of anisotropic Besov spaces[J]. Constr. Approx., 2010, 31(2): 149-194. DOI:10.1007/s00365-009-9058-y
[5]
Dekel S, Petrushev P, Weissblat T. Hardy spaces on $ {\mathbb{R}}^n$ with pointwise variable anisotropy[J]. J. Fourier Anal. Appl., 2011, 17(5): 1066-1107. DOI:10.1007/s00041-011-9176-3
[6]
Dekel S, Weissblat T. On dual spaces of anisotropic Hardy spaces[J]. Math. Nachr., 2012, 285(17-18): 2078-2092. DOI:10.1002/mana.201100329
[7]
Almeida V, Betancor J, Rodríguez-Mesa L. Molecules associated to Hardy spaces with pointwise variable anisotropy[J]. Inter. Equ. Oper. Theory, 2017, 89(3): 301-313. DOI:10.1007/s00020-017-2403-9
[8]
Calderón A P, Torchinsky A. Parabolic maximal functions associated with a distribution[J]. Adv. Math, 1975, 16(1): 1-64. DOI:10.1016/0001-8708(75)90099-7
[9]
Bownik M. Anisotropic Hardy spaces and wavelets[J]. Mem. Amer. Math. Soc., 2003, 164(781): 1-122.