A super wavelet of length n is an n-tuple (ψ 1,ψ 2,…,ψ n ) in the product space Õj=1n L2(mathbbR)prod_{j=1}^{n} L^{2}(mathbb{R}), such that the coordinated dilates of all its coordinated translates form an orthonormal basis for Õj=1n L2 (mathbbR)prod_{j=1}^{n} L^{2} (mathbb{R}). This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η 1,η 2,…,η n ) in Õj=1nL2 (mathbbR)prod_{j=1}^{n}L^{2} (mathbb{R}) such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for Õj=1n L2(mathbbR)prod_{j=1}^{n} L^{2}(mathbb{R}). In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,…,E m ) is said to be τ-disjoint if the E j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E 1,E 2,…,E m ) of frame sets (i.e., η j defined by [^(hj)]=frac1Ö{2p}cEjwidehat{eta_{j}}=frac{1}{sqrt{2pi}}chi_{E_{j}} is a frame wavelet for L 2(ℝ) for each j) lead to a super frame wavelet (η 1,η 2,…,η m ) for Õj=1m L2 (mathbbR)prod_{j=1}^{m} L^{2} (mathbb{R}) where [^(hj)]=frac1Ö{2p}cEjwidehat{eta_{j}}=frac{1}{sqrt{2pi}}chi_{E_{j}}. In the case of super tight frame wavelets, we prove that (η 1,η 2,…,η m ), defined by [^(hj)]=frac1Ö{2p}cEjwidehat{eta_{j}}=frac{1}{sqrt{2pi}}chi_{E_{j}}, is a super tight frame wavelet for ∏1≤j≤m L 2(ℝ) with frame bound k 0 if and only if each η j is a tight frame wavelet for L 2(ℝ) with frame bound k 0 and that (E 1,E 2,…,E m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏1≤j≤m L 2(ℝ) by mathfrakS(m)mathfrak{S}(m) and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ∏1≤j≤m L 2(ℝ) by mathfrakSk0(m)mathfrak{S}^{k_{0}}(m). We further prove that mathfrakS(m)mathfrak{S}(m) and mathfrakSk0(m)mathfrak{S}^{k_{0}}(m) are both path-connected under the ∏1≤j≤m L 2(ℝ) norm, for any given positive integers m and k 0.
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机译：长度为n的超小波在乘积空间Õ<中是n元组（ψ 1 ，ψ 2 ，…，ψ n ）。 sub> j = 1 n L 2 （mathbbR）prod_ {j = 1} ^ {n} L ^ {2}（mathbb {R}） ，使得所有协调翻译的协调扩张都形成Õ j = 1 n L 2 （mathbbR）prod_ {j的正交基础= 1} ^ {n} L ^ {2}（mathbb {R}）。该概念被推广到所谓的超帧小波，超紧帧小波和超归一化紧帧小波（或超级Parseval帧小波），即n元组（η 1 ，η j = 1 n L 2 （mathbbR）中的> 2 ，…，η n ） ）prod_ {j = 1} ^ {n} L ^ {2}（mathbb {R}），以便其所有协调平移的协调扩张形成Õ的框架，紧框架或归一化紧框架j = 1 n L 2 （mathbbR）prod_ {j = 1} ^ {n} L ^ {2}（mathbb {R}）。在本文中，我们研究了由设置的理论函数（称为s元素框架小波）定义了傅里叶变换的超框架小波和超紧框架小波。如果E 1 ，E 2 ，…，E m 的m个元组被称为τ不相交。 sub> j 在2π平移下成对不相交。我们证明帧集合（即η）的τ不相交的m元组（E 1 ，E 2 ，…，E m ）由[^（h j ）] =frac1Ö{2p} c E j widehat {eta_ {定义的 j j}} = frac {1} {sqrt {2pi}} chi_ {E_ {j}}是每个j的L 2 （ℝ）的帧小波，导致超帧小波（η sub j = 1 m 1 ，η 2 ，…，η m ） > L 2 （mathbbR）prod_ {j = 1} ^ {m} L ^ {2}（mathbb {R}）其中[^（h j ）] = frac1Ö{2p} c E j widehat {eta_ {j}} = frac {1} {sqrt {2pi}} chi_ {E_ {j}}。对于超紧框架子波，我们证明（η 1 ，η 2 ，…，η m ）由[^ （h j ）] =frac1Ö{2p} c E j widehat {eta_ {j}} = frac {1} {sqrt {2pi }} chi_ {E_ {j}}是∏ 1≤j≤m L 2 （ℝ）的超紧帧子波，其帧边界为k 0 当且仅当每个η j 是L 2 （ℝ）的紧帧小波且帧绑定k 0 且（E 1 ，E 2 ，…，E m ）是τ不相交的。用mathfrakS（m）mathfrak {S}（）表示∏ 1≤j≤m L 2 （ℝ）的所有τ不相交s元超帧小波的集合m）和∏ 1≤j≤m L 2 <的所有s元素超紧帧小波（具有相同的帧边界k 0 ）的集合/ sup>（ℝ）作者mathfrakS k 0 （m）mathfrak {S} ^ {k_ {0}}（m）。我们进一步证明mathfrakS（m）mathfrak {S}（m）和mathfrakS k 0 （m）mathfrak {S} ^ {k_ {0}}（m）对于任何给定的正整数m和k 0 ，都在∏ 1≤j≤m L 2 （ℝ）范数下进行路径连接。
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