Narrow Results

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τ_i, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τ_i is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τ_i to operators on the Hilbert space L²(μ). Instead, we show that it is possible to realize the maps τ_i quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫_X|φ|² dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and

We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

This paper examines two properties of refinable functions. The first provides conditions under which a refinable function is strictly positive and the other presents a conjecture about cardinal interpolation by refinable functions. The conjecture is supported by some analysis and also computation.

This paper is devoted to the construction of interpolating multiresolutions using Lagrange polynomials and incorporating a position dependency. It uses the Harten's framework²¹ and its connection to subdivision schemes. Convergence is first emphasized. Then, plugging the various ingredients into the wavelet multiresolution analysis machinery, the construction leads to position-dependent interpolating bases and multi-scale decompositions that are useful in many instances where classical translation-invariant frameworks fail. A multivariate generalization is proposed and analyzed. We investigate applications to the reduction of the so-called Gibbs phenomenon for the approximation of locally discontinuous functions and to the improvement of the compression of locally discontinuous 1D signals. Some applications to image decomposition are finally presented.

In this work, we give an introduction of uniform algebraic hyperbolic $B$-splines (UAH $B$-splines) generated over the space spanned by $\{sinh(\alpha x),cosh(\alpha x),\dots ,x^{k-3}sinh(\alpha x),x^{k-3}cosh(\alpha x)\}$, in which $k$ is an integer larger than or equal to $3$ and $\alpha$ is a tension parameter. Then, we construct a general formula of the refinement equation for any given order $k$ of these $B$-splines. Using the matrix version of this equation, we have also constructed the subdivision formula for UAH $B$-spline curves. In order to introduce a new reverse subdivision framework, entitled “Smooth Reverse Subdivision” associated with the cubic UAH $B$-splines, by continuing this process, we present a new multiresolution technique for general topology curves. We illustrate our results by numerical experiments.

Integral equations play a vital role in the fields of applied and computational mathematics. This paper introduces a unique subdivision-based collocation technique for solving the initial value problems (IVPs) arising from Volterra integral equations (VIEs). A suitable subdivision scheme is employed to develop the numerical algorithm for solving the VIEs. Several numerical illustrations are tested and compared with the existing methods for assessing the efficiency of the proposed numerical algorithm and error estimation.