Ondelettes, analyses multirésolutions et filtres miroirs en quadrature. (Wavelets, multiscale analysis and quadrature mirror filters) (Q1173890)

Let \(\{V_ j\}_{j\in\mathbb{Z}}\) be a multiresolution analysis in \(L^ 2(\mathbb{R}^ n)\), and let \(\phi\) be a function such that \(\{\phi(x- k)\}_{k\in\mathbb{Z}^ n}\) forms an orthonormal base of \(V_ 0\). If \(\phi\) satisfies \((1+| x|)^ m\phi(x)\in L^ 2(\mathbb{R}^ n)\) for all \(m\in\mathbb{N}\), or equivalently \(\hat \phi(x)\in H^ m(\mathbb{R}^ n)\) for all \(m\in\mathbb{N}\), then \(\{V_ j\}_{j\in\mathbb{Z}}\) is called an analysis with regular filter. The author solves the following problem. Let \(\phi\) be a function such that \(\hat\phi(x)\in H^ m(\mathbb{R}^ n)\) for all \(m\in\mathbb{N}\) and such that \(\{\phi(x-k)\}_{k\in\mathbb{Z}^ n}\) is an orthonormal sequence. Under what conditions does the sequence of the Hilbert spaces \(V_ j\) generated by \(\{2^{nj/2}\phi(2^ jx-k)\}_{k\in\mathbb{Z}^ n}\) form a multiresolution analysis? A crucial role is played by the function \(m_ 0(x)\) defined by \(\hat\phi(2x)=m_ 0(x)\hat\phi(x)\). Next, the author establishes a necessary and sufficient condition in order that a function \(m_ 0(x)\) generates an analysis with regular filter.