In this paper, we study approximate duals of $g$-frames and fusion frames in Hilbert $C^ast-$modules. We get some relations between approximate duals of $g$-frames and biorthogonal Bessel sequences, and using these relations, some results for approximate duals of modular Riesz bases and fusion frames are obtained. Moreover, we generalize the concept of $Q-$approximate duality of $g$-frames and ...

Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to c...

There exist efficient methods to compute the wavelet coefficients of a function f(t) from its point samples f ( T [n + τ ] ) , n ∈ N. However, in many applications the available samples are average samples of the type ∫∞ −∞ f ( T [t + n + τ ] ) u(t)dt, where the averaging function u(t) reflects the characteristic of the acquisition device. In this work, methods to compute the coefficients in a ...

Interpolatory quadrature rules whose abscissas are zeros of a biorthogonal polynomial have proved to be useful, especially in numerical integration of singular integrands. However, the positivity of their weights has remained an open question, in some cases, since 1980. We present a general criterion for this positivity. As a consequence, we establish positivity of the weights in a quadrature r...

Digital image compression with multiresolution singular value decomposition is compared with discrete cosine transform, discrete 9/7 biorthogonal wavelet transform, Karhunen–Loève transform, and combinations thereof. The coding methods used SPIHT and run-length with Huffmann coding. The performances of these methods differ little from each other. Generally, the 9/7 biorthogonal wavelet transfor...

In this paper we connect the design of biorthogonal modulated filter banks with the idea of lifting schemes, which were proposed for the construction of biorthogonal wavelets. Based on this formulation, we derive a lattice-like structure for the polyphase components. As the rotationbased structures for the design and implementation of paraunitary modulated filter banks, our structure automatica...

This paper deals with the construction of biorthogonal p -wavelet packets on R+ related to the Walsh polynomials and their properties are investigated by means of Walsh-Fourier transform. Three biorthogonal formulas regarding these p -wavelet packets are derived. Moreover, it is shown how to obtain several new Riesz bases of the space L2(R+) by constructing a series of subspaces of these p -wav...

We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R+ such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R+. Such wavelets are suitable to study the Navier–Stokes equation...

In this paper, by combining the ideas of the recursive wavelets with second-generation wavelets, a family of recursive biorthogonal interpolating wavelets (RBIWs) is developed. RBIW’s have simple shape parameter vectors on each level, which allows a multichannel decomposition algorithm and provides a flexible structure for designing signal-adapted interpolating filter banks. In the single-level...

In this paper, the theory and design of a class of perfectreconstruction (PR) biorthogonal nonuniform cosine-modulated filter banks (CMFB) is proposed. It is based on a recombination or merging structure previously proposed by the authors [ 13. By relaxing the original CMFB and the recombination transmultiplexer (TMUX) to be biorthogonal. nonuniform CMFBs with lower system delay can be obtained...