In this paper we survey some recent results on cascade algorithms. Let a be a finitely supported sequence on ZZ. The cascade operator Qa is the the linear operator on Lp(IR) (1 ≤ p ≤ ∞) given by Qaf := ∑ j∈Z a(j)f(2 · − j), f ∈ Lp(IR). The iteration scheme Qaf (n = 1, 2, . . .) is called the cascade algorithm associated with a. The Lp convergence of a cascade algorithm is characterized in terms...

Multiple InputMultiple Output (MIMO) biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper we first derive some theoretical properties of MIMO biorthogonal partners. We develop conditions for the existence of MIMO biorth...

A biorthogonal wavelet is derived from a pair of biorthogonal refinable functions using the standard technique in multiresolution analysis. In this paper, we introduce the concept of projectable refinable functions and demonstrate that many multivariate refinable functions are projectable; that is, they essentially carry the tensor product (separable) structure though themselves may be non-tens...

In this paper, a new numerical method for solving fractional optimal control problems (FOCPs) is presented. The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon biorthogonal cubic Hermite spline multiwavelets approximations. The properties of biorthogonal multiwavelets are first given. The operational matrix of fractional Riemann-Lioville in...

—This paper proposes new concepts of the windowed all phase biorthogonal transform (WAPBT), which is inspired by the all phase biorthogonal transform (APBT). In the light of windowed all phase digital filter theory, windowed all phase biorthogonal transforms is proposed. The matrices of WAPBT based on DFT, WHT, DCT and IDCT are deduced, which can be used in image compression instead of the con...

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast ...

This paper presents a construction of compactly supported biorthogonal spline wavelets in L2(IR ). In particular, a concrete method for the construction of bivariate compactly supported biorthogonal wavelets from box splines of increasing smoothness is provided. Several examples are given to illustrate the method. Key-Words:multivariate biorthogonal wavelets, multivariate wavelets, box splines,...

Biorthogonal bases of compactly-supported wavelets are characterized by the FIR perfect-reconstruction filterbanks to which they correspond. In this paper we develop explicit representations of all such filterbanks, allowing us to generate every possible biorthogonal compactly-supported wavelet basis. For these filterbanks, the product H(z) = H(z) e H(z) of the two lowpass filters must have N 2...

In this paper, the properties of the all phase biorthogonal transform (APBT) matrix are deduced based on the concepts of all phase biorthogonal transform. A novel algorithm is presented to compress color image using the all phase Walsh biorthogonal transform (APWBT), the all phase discrete cosine biorthogonal transform (APDCBT) and the all phase inverse discrete cosine biorthogona transform (AP...

We construct oblique multi-wavelets bases which encompass the orthogonal multi-wavelets and the biorthogonal uni-wavelets of Cohen, Deaubechies and Feauveau. These oblique multi-wavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of wavelet theory to accommodate a wider variety of wavelet shapes and properties. Moreover, oblique multi-wavelets ca...