A suucient condition for the orthonormality of reenable vector functions is given. Then we construct examples of orthonormal compactly supported multiscal-ing functions and their associated multiwavelets with arbitrarily high regularity in the univariate and bivariate settings. We use this suucient condition to check their orthonor-mality.

In this paper, model sets for continuous–time linear time invariant systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalise the well known Laguerre and two–parameter Kautz cases. It is shown that the obtained model sets are norm dense in the Hardy space H1(Π) under the same condition as previously derived by the authors for the norm denseness in the (Π ...

Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M , we demonstrate in a constructive way that we can construct compactly supported tight M -wavelet frames and orthonormal M -wavelet bases in L2(R) of exponential decay, which are derived from compactly supported M -refinable functions,...

Let wρ x : |x|exp −Q x , ρ > −1/2, where Q ∈ C2 : −∞,∞ → 0,∞ is an even function. In 2008 we have a relation of the orthonormal polynomial pn w2 ρ;x with respect to the weightw 2 ρ x ; p′ n x An x pn−1 x − Bn x pn x − 2ρnpn x /x, where An x and Bn x are some integrating functions for orthonormal polynomials pn w2 ρ;x . In this paper, we get estimates of the higher derivatives of An x and Bn x ,...

Square-integrable functions f ∈ L are those of length ‖f‖L2 = 〈f, f〉 L2 < ∞. A subset D ⊂ L is said to be dense, if any f ∈ L can be approximated by a sequence fn ∈ D. This means limn ‖f − fn‖L2 = 0. An orthonormal basis {Ωn} is a set of orthonormal vectors whose finite linear combinations are dense. A linear transformation T is continuous on L if ‖Tf‖L2 ≤ M‖f‖L2 for some constant M < ∞. A cont...

This paper studies continuous-time system model sets that are spanned by "xed pole orthonormal bases. The nature of these bases is such as to generalise the well-known Laguerre and two-parameter Kautz bases. The contribution of the paper is to establish that the obtained model sets are complete in all of the Hardy spaces H p (P), 1(p(R, and the right half plane algebra A(P) provided that a mild...

Many nonlinear systems can be described by a Wiener-Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs). The nonlinearity is modeled by a multivariate polynomial. In general, an infinite number of OBFs is needed for an exact representation of the system. This paper considers the approximation of a Wiener system with finite-order infin...

One approach to the problem lies in showing that expansion (1) is equivalent to another expansion, usually a Fourier series, whose behavior is known. In the case of orthogonal polynomials, where un{x) =x ~, the writer was able to show, under conditions not too restrictive, that the expansions of a function in terms of the two sets of orthogonal polynomials corresponding, respectively, to differ...

There has recently been interest in the use of orthonor-mal bases for the purposes of SISO system identiication. Concurrently, but separately, there has also been vigorous work on estimation of MIMO systems by computa-tionally cheap and reliable schemes. These latter ideas have collectively become known as`4SID' methods. This paper is a contribution overlapping these two schools of thought by s...

We construct a new orthonormal basis for L(R), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L(R) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L(R). Orthonormal ridgelet expansions have an interesting application in nonlinear app...