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A characterization of dimension functions of wavelets. (English) Zbl 0979.42018
A finite set \(\Psi = \{ \psi^1, \ldots, \psi^L \} \subset L^2(R^d)\) is called an orthonormal multiwavelet, if the collection \[ \{ \psi^l_{j,k} (x) = |\det A|^{j/2} \psi^l (A^j x-k) : j\in Z,\;k \in Z^d,\;l=1, \ldots, L \} \] forms an orthonormal basis for \(L^2(R^d)\). Here, \(A\) is an expansive matrix which preserves \(Z^d\). The dimension function of such a multiwavelet is defined as \[ D_\Psi (\xi) = \sum_{l=1}^L \sum_{j=1}^\infty \sum_{k\in Z^d} |\widehat \psi^l (B^j(\xi+k))|^2, \] where \(B=A^T\).
It is well known that dimension functions are measurable, nonnegative, \(Z^d\)-periodic and integer-valued. In this paper the authors provide a full characterization of all dimension functions, by adding to the above properties the following conditions: \[ \int_{(-1/2,1/2]^d} D(\xi) d\xi = L/(q-1), \quad \text{where }q = |\det A|,\tag{1} \]
\[ \liminf_{n\to \infty} D(B^{-n}\xi) \geq 1,\tag{2} \]
\[ \sum_{k \in Z^d} \chi_\Delta(\xi+k) \geq D(\xi), \text{ a.e. for }\Delta = \{ \xi \in R^d: D(B^{-j}\xi) \geq 1;\;j\in N\cup\{0\} \},\tag{3} \]
\[ \sum_{d\in {\mathcal D}} D(\xi+B^{-1}d) = D(B\xi)+L, \quad \text{a.e.} .\tag{4} \] Here, \({\mathcal D}= \{d_1, \dots, d_q \}\) denotes the set of representatives of \(Z^d/BZ^d\), with \(d_1=0\).
The authors further study the properties of the dimension function using these characterizing conditions, and provide several examples of wavelets for given dimension functions. In particular, they show that for any dimension function \(D\), there exists an MSF wavelet whose dimension function is \(D\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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