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A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L ^2(ℝ^n) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B . Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝ^n. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce “long and narrow” window functions, with various orientations, well-suited to applications in image processing.

This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Papoulis, could be carried over in many cases to the spline spaces and other shift-invariant spaces. Papoulis’ method is based on the sampling output of linear, time-invariant systems. Unser and Zerubia formalized Papoulis’ approach in the context of shift-invariant spaces. However, it is not easy to provide useful conditions under which the Unser-Zerubia criterion provides convergent and stable sampling expansions. Here we review several methods for validating the Unser-Zerubia approach for periodic nonuniform sampling, which is a very special case of generalized sampling. The Zak transform plays an important role.

We give a complete solution to the problem of sampling and interpolation of functions in PW _σ( R ) on finite unions of shifted lattices in R of the form $$ \Lambda j = \frac{1} {{2\sigma _j }}Z + \alpha _j ,j = 1, \ldots ,m $$ where σ_ j > 0, α_ j ∈ R , and ∑_ j σ_ j =σ. At points where more than one lattice intersect, we sample the function and its derivatives. None of the results or techniques employed is new, but a systematic and elementary treatment of this situation does not seem to exist in the literature. Sampling on unions of shifted lattices includes classical sampling, bunched or periodic sampling, and sampling with derivatives. Such sampling sets arise in deconvolution, tomography, and in the theory of functions bandlimited to convex regions in the plane.

In this chapter we discuss the problem of finding the shift-invariant space model that best fits a given class of observed data F . If the data is known to belong to a fixed—but unknown—shift-invariant space V ( Φ ) generated by a vector function Φ , then we can probe the data F to find out whether the data is sufficiently rich for determining the shift-invariant space. If it is determined that the data is not sufficient to find the underlying shift-invariant space V , then we need to acquire more data. If we cannot acquire more data, then instead we can determine a shift-invariant subspace S ⊂ V whose elements are generated by the data. For the case where the observed data is corrupted by noise, or the data does not belong to a shift-invariant space V ( Φ ), then we can determine a space V ( Φ ) that fits the data in some optimal way. This latter case is more realistic and can be useful in applications, e.g., finding a shift-invariant space with a small number of generators that describes the class of chest X-rays.

A description of the fine structure of a refinable, shift-invariant sub-space of L ^2(ℝ) is presented. This fine structure is exhibited through the existence of a canonical frame of functions in such a space, and a related notion of frequency content in these frame elements uniquely determines a multiplicity function that quantifies a redundancy of the frequencies. The refinability of the subspace can then be described by a pair of matrices of periodic functions that satisfy a set of equations, related to the multiplicity function, which play the role of high-dimensional filter equations.

For a separated sequence Λ={λ_ k }_ k∈z of real numbers there is a close link between the lower and upper densities D ^−( Λ ), D ^+( Λ ) and the frame properties of the exponentials $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}:} $$ in fact, $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}} $$ is a frame for its closed linear span in L ^2(−ν, ν) for any ν ∈ (0, π D ^-(Λ)) ∪ (π D ^+(Λ),∞) . We consider a classical example presented already by Levinson [ 11 ] with D ^-(Λ) = D ^+(Λ) = 1; in this case, the frame property is guaranteed for all ν ∈ (0; ∞) ∖ { π }. We prove that the frame property actually breaks down for ν = π . Motivated by this example, it is natural to ask whether the frame property can break down on an interval if D ^−( Λ ) ≠ D ^+( Λ ). The answer is yes: We present an example of a family Λ with D ^−( Λ ) ≠ D ^+( Λ ) for which $$ \{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}} $$ has no frame property in L ^2(−ν, ν) for any ν ∈ ( π D ^−( Λ ), π D ^+( Λ )).