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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.

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Abstract. We consider solutions of a system of refinement equations written in the form φ = α∈Z a(α)φ(2 · −α), where the vector of functions φ = (φ 1 , . . . , φ r ) T is in (Lp(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 Qa defined on (Lp(R)) r by Qaf := 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 n a f ) n=1,2,... in the Lp-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.

In this paper we consider functional equations of the form . , φ r )T is an r × 1 vector of functions on the s-dimensional Euclidean space, a(α), α ∈ Z s , is a finitely supported sequence of r × r complex matrices, and M is an s ×s isotropic integer matrix such that lim n→∞ M −n = 0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φ j , j = 1, . . . , r, belonging to the Sobolev space W k p (R s )? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Q a associated with the sequence a is defined byLet 0 be a nontrivial r × 1 vector of compactly supported functions in W k p (R s ). The iteration scheme n = Q a n−1 , n = 1, 2, . . . , is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. VECTOR SUBDIVISION SCHEMES IN SOBOLEV SPACES 129The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang-Fix type conditions that determine the feasible set of initial vectors.  2002 Elsevier Science

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈Zs∖{0}]. Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces. A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.