Generalized wavelets and compactly supported scaling functions
This paper solves two old problems in wavelet theory concerning the relation between multiresolution analyses and wavelet bases in dimension 1 with a general dilation factor \(A\), \(A \in Z\), \(A \geq 2\). A ``multiresolution analysis'' (MRA in short) with respect to a dilation \(A\) is an increasing sequence \((V_ j)_{j \in Z}\) of closed subspaces of \(L^ 2(R)\) with the following properties: (1) \(\bigcup_{j \in Z}V_ j\) is dense in \(L^ 2(R)\) and \(\bigcap_{j \in Z}V_ j=\{0\}\), (2) \(f(x) \in V_ j \to f(Ax) \in V_{j+1}\) for all \(j \in Z\), (3) \(V_ 0\) possesses an orthonormal basis of the form \(\bigl\{ \varphi(x- k)\), \(k \in Z \bigr\}\). \(\varphi\) is called the scaling function of the MRA. A ``wavelet basis'' generated by the wavelets \(\psi_ 1,\ldots, \psi_{A-1}\) is an orthonormal basis of the form \(\bigl\{ A^{j/2} \psi_ r (A^ jx-k),\;j,k \in Z,\;r=1,\ldots,A-1\bigr\}\). For the construction of a wavelet basis one considers the orthogonal complement \(W_ 0\) of \(V_ 0\) in \(V_ 1\) and shows that there exist wavelets \(\psi_ 1,\ldots,\psi_{A-1}\), such that the integer translates of the \(\psi_ r\)'s form an orthonormal basis of \(W_ 0\). It is known that these wavelets can be chosen to have the same smoothness and decay properties as the scaling function \(\varphi\), but for \(A>2\) no construction of compactly supported wavelets had been obtained. All interesting wavelet bases have been derived from a MRA following this strategy. Therefore it had been conjectured since the beginnings of wavelet theory that this is the only possible construction for wavelet basis with sufficient smoothness and decay properties. This paper solves both problems and provides several versions of the relation between MRA's and wavelet bases. Theorem 1: If \((V_ j)_{j \in Z}\) is a MRA with dilation factor \(A \geq 2\), \(A \in Z\), and with a compactly supported scaling function \(\varphi\), then there exist \(A-1\) compactly supported wavelets \(\psi_ r\) so that \(\bigl\{ \psi_ r(x-k),\;k \in Z,\;r=1,\ldots,A-1\bigr\}\) is an orthonormal basis for \(W_ 0\). Then the collection \(\bigl\{ A^{j/2}\psi_ r(A^ jx-k),\) \(j,k \in Z,\;r=1,\ldots,A-1\bigr\}\) is an orthonormal basis of \(L^ 2(R)\). If \(\varphi\) is Hölder continuous in \(C^ \alpha\), then also \(\psi_ r \in C^ \alpha\) for \(r=1,\ldots,A- 1\). Moreover, the length of the support of the \(\psi_ r\)'s is at most the length of supp \(\varphi\). Theorem 2: Conversely, if \(\bigl\{ A^{j/2} \psi_ r(A^ jx-k)\), \(j,k \in Z,\;r=1,\ldots,A-1\bigr\}\) is an orthonormal basis of \(L^ 2(R)\) with continuous and compactly supported wavelets \(\psi_ r\), then \(\bigl\{ \psi_ r(x-k),\;r=1,\ldots,A-1\bigr\}\) is an orthonormal basis for a subspace \(W_ 0\) associated to a MRA with a compactly supported scaling function \(\varphi\). \(\varphi\) is continuous and its support is smaller than the maximum support of the \(\psi_ r\)'s. If \(\psi_ r \in C^ \alpha\) for \(r=1,\ldots,A-1\), then also \(\varphi \in C^ \alpha\). The proof involves a detailed and profound investigation of closed subspaces of \(L^ 2(R)\) which contain compactly supported functions and which are invariant under translations by integers. As a result of independent interest, it is shown by a very elegant optimization argument that any such space contains an orthonormal set of the form \(\bigl\{ \varphi(x-k),\;k \in Z\bigr\}\) with compactly supported \(\varphi\). The paper contains several variations and extensions of these two theorems. a) A version of the theorem holds, if \(V_ 0\) is generated by several scaling functions of compact support. b) The conclusions of Theorems 1 and 2 remain true if the assumption of ``compact support'' is replaced by ``exponential decay'' of \(\varphi\) or of \(\psi_ r\). The proof requires completely different methods. c) Finally, similar results are shown for biorthogonal wavelet bases. The paper contains many new ideas and techniques and it is clearly one of the most interesting and profound works in wavelet theory so far.
scientific article; zbMATH DE number 404302