Generalized sampling in a shift invariant subspace V of L 2(ℝ) is considered. A function f in V is processed with different filters L m and then one tries to reconstruct f from the samples L mf(j′k). We develop a theory of how to do this in the case when V possesses a shift invariant frame. Special attention is paid to the question: How to obtain dual frames with compact support?

An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function ϕ with Fourier transform equal to the convolution of g with the characteristic function living on the interval [-pi,pi]. We explain why analysis of this particular generating function can be more likely to provide large jitter bounds ε such that any f ∈ V can be reconstructed from perturbed integer samples f(k + ε_k) whenever the supremum of |ε_k| is smaller than ε. We use this natural deconvolution to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of ϕ for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces.

We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find epsilon so that given perturbations (lambda(k)) satisfying sup vertical bar lambda(k)vertical bar < epsilon, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k+lambda(k)). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits epsilon for the reconstruction. We show how it works in two concrete situations.

A subspace V of L2(ℝ) is called shift-invariant if it is the closed linear span of integer-shifted copies of a single function.As a complement to classical analysis techniques for sampling in such spaces, we propose a method which is based on a simple interpolation estimate of a certain coefficient mapping. Then we use this method to derive both new results and relatively simple proofs of some previously known results. Among these are some results of rather general nature and some more specialized results for B-spline wavelets. The main problem under study is to find a shift x0 and an upper bound δ such that any function f ∈ V can be reconstructed from a sequence of sample values (f(x0 + k + δk))k∈ℤ, either when all δk = 0 or in the irregular sampling case with an upper bound supk|δk| < δ.

A wavelet prefilter maps sample values of an analyzed signal to the scaling function coefficient input of standard discrete wavelet transform (DWT) algorithms. The prefilter is the inverse of a certain postfilter convolution matrix consisting of integer sample values of a noninteger-shifted wavelet scaling function. For the prefilter and the DWT algorithms to have similar computational complexity, it is often necessary to use a "short enough" approximation of the prefilter. In addition to well-known quadrature formula and identity matrix prefilter approximations, we propose a Neumann series approximation, which is a band matrix truncation of the optimal prefilter, and derive simple formulas for the operator norm approximation error. This error shows a dramatic dependence on how the postfilter noninteger shift is chosen. We explain the meaning of this shift in practical applications, describe how to choose it, and plot optimally shifted prefilter approximation errors for 95 different Daubechies, Symlet, and B-spline wavelets. Whereas the truncated inverse is overall superior, the Neumann filters are by far the easiest ones to compute, and for some short support wavelets, they also give the smallest approximation error. For example, for Daubechies 1-5 wavelets, the simplest Neumann prefilter provide an approximation error reduction corresponding to 100-10 000 times oversampling in a nonprefiltered system.

In this paper we derive and compare several different vibration analysis techniques for automatic detection of local defects in bearings. Based on a signal model and a discussion on to what extent a good bearing monitoring method should trust it, we present several analysis tools for bearing condition monitoring and conclude that wavelets are especially well suited for this task. Then we describe a large-scale evaluation of several different automatic bearing monitoring methods using 103 laboratory and industrial environment test signals for which the true condition of the bearing is known from visual inspection. We describe the four best performing methods in detail (two wavelet-based, and two based on envelope and periodisation techniques). In our basic implementation, without using historical data or adapting the methods to (roughly) known machine or signal parameters, the four best methods had 9–13% error rate and are all good candidates for further fine-tuning and optimisation. Especially for the wavelet-based methods, there are several potentially performance improving additions, which we finally summarise into a guiding list of suggestion.

We prove some reiteration formulas for the Cobos-Peetre polygon method for $n+1$ tuples that consists of spaces $A_i$ where $A_i$ is of class $\theta_i$ with respect to a compatible pair $(X,Y)$. If $\theta_i$ is suitably chosen, the $J$- and $K$-method coincides and is equal to a space $(X,Y)_{\nu,q}$. For arbitrary chosen $\theta_i$ the $J$- and $K$-spaces will not, in general, coincide. In particular, we show that interpolation of Lorentz spaces over the unit square yields that the $K$-space is the sum of two Lorentz spaces whereas the $J$-space is the intersection of the same two Lorentz spaces.

We prove some equivalence formulas for the K functional for three spaces. In particular we calculate K(t, s, f, L_p0, L_p1, L_p2). We also give an integral formula connecting the K functional for triples of the type (A₀, A₁, A₂), where A₁ is of class C(θ, A₀,A₂), with the K functional for the pair (A₀, A₂). This formula is used to prove a reiteration formula for interpolation of the triple (A₀, A_l, A₂) when Ai is of class C(θ_i, X, Y)

We give necessary and sufficient conditions on a general cone of positive functions to satisfy the Decomposition Property (DP) introduced in [5] and connect the results with the theory of interpolation of cones introduced by Sagher [9]. One of our main result states that if Q satisfies DP or equivalently is divisible, then for the quasi-normed spaces E0 and E1, According to this formula, it yields that the interpolation theory for divisible cones can be easily obtained from the classical theory