Sherman Riemenschneider

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.

Wavelets and pre-wavelets in low dimensions

Abstract In Riemenschneider and Shen (in “Approximation Theory and Functional Analysis” (C. K. Chui, Ed.), pp. 133–149, Academic Press, New York, 1991 ) an explicit orthonormal basis of wavelets for L2( R s), s=1,2,3, was constructed from a multiresolution approximation given by box splines. In other words, L2( R s) has the orthogonal decomposition ⊕ Wν. (∗) ν ϵ Z Orthonormal bases for the spaces Wν, are given by {2 νs 2 K μ (2 ν · −j)} , j∈ Z s, μ ϵ Z 2s⧹ 0, where Z s2 and the “wavelets” Kμ are 2s − 1 cardinal splines with exponential decay. In this paper, we consider multiresolutions generated by suitable compactly supported and symmetric functions ϑ and explicitly construct 2s − 1 compactly supported functions ϑμ, μ ϵ Z 2s⧹ 0, such that the translates ϑμ(· − j), j∈ Z s, are an unconditional basis for W0. Thus, the functions ϑμ(2ν· − j), ν ϵ Z, j∈ Z s, μ ϵ Z 2s ⧹ 0 comprise a basis for the orthogonal decomposition (∗) (the functions are orthogonal for different ν because the decomposition is orthogonal, but neither the translates nor the functions will be orthogonal for given ν). The functions are given as ϑ μ (·/2) 2 s = μ ∗′ β μ with the sequences βμ formed from a single sequence by translation and change in sign pattern. We also discuss various ways to regain some of the orthogonality lost by requiring compact support.

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form formula math where the vector of functions Φ = (Φ 1 ,...,Φ r ) T is in (L p (R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Q a defined on (L p (R)) r by Q a f:= Σ α ∈ z a(α)f(2.-α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q a n f) n=1,2... in the L p -norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L 2 -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

Smoothness of Multiple Refinable Functions and Multiple Wavelets

We consider the smoothness of solutions of a system of refinement equations written in the form

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

\phi = \sum\nolimits_{\ga\in\ZZ} a(\ga)\phi({2\,\cdot}-\ga),

Approximation by multiple refinable functions

where the vector of functions

Weak Interpolation in Banach Spaces

\phi=(\phi_1,\ldots,\phi_r)^T

Construction of compactly supported biorthogonal wavelets: II

is in