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
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.
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.,
where $ M_\xi $ is an invertible matrix and $ c_\xi $ is the center.
Definition 2.1 We say that
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
$ {\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
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
Proof For any $ y\in\mathbb{B}^n $ and $ t<0 $, by (2.2), we obtain
which implies that
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
and, for each measurable function $ f $,
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
$ \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 $,
Then
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} $:
Obviously, $ \|M_{x, 0}\| = 1 $.
$ {\rm{(ii)}} $ The one-parameter family of diagonal dilation matrices
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,
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
Obviously, $ \|M_{x, 0}\| = \|P^{-1}\| $.
$ {\rm{(v)}} $ Lighted by [4, Theorem 7.3], we give a concrete example for the ellipsoid cover $ \Theta $:
with
where
Here, $ M_{x, \, t} = \text{diag}(\sigma_1, \, \sigma_2) $. Obviously, $ \|M_{x, \, 0}\| = \|\text{I}_{n\times n}\| = 1 $,
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, $
From this, we deduce that, for any $ x\in \text{supp} f, $
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
From this and (2.2), for any $ x\in N\mathbb{B}^n $, $ y\in \mathbb{R}^n $ and $ t>0 $, we deduce that
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 $,
which, together with (3.1), implies that
$ {\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
Since $ \text{supp} f\subset N\mathbb{B}^n $, we only need to estimate the above integral under the following condition:
For any $ t>0 $, by this, (2.2) and (2.4), we have
and hence $ x\in (Ca_5+N)\mathbb{B}^n $. From this, we further deduce that
By this, (3.2), Fatou's lemma, the uniformly continuity of $ f $ on $ (Ca_5+N)\mathbb{B}^n $ and the fact that
we have
This finishes the proof of Theorem 2.4(ⅱ).