Wavelet frames and duals: a short survey

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

We consider weighted function spaces of Sobolev-Besov type and Schrodinger type operators on noncompact Riemannian manifolds with bounded geometry. First we give characterization of the spaces in terms of wavelet frames. Then we describe the necessary and sufficient conditions for the compactness of Sobolev embeddings between the spaces. An asymptotic behavior of the corresponding entropy numbers is calculated. At the end we use the asymptotic behavior to estimate the number of negative eigenvalues of the Schrodinger type operators.

We give an expression for the most general input–output map associated with the members of a certain important large family of multidimensional linear shift‐invariant systems with bounded Lebesgue‐measurable inputs. The expression given is an iterated function‐space limit of a convolution. We also give a necessary and sufficient condition under which the limit can be written as a convolution with an integrable impulse–response function. A key role is played by a certain family of weighting operators. It is observed that for the large family of inputs and maps addressed, the Dirac impulse–response concept is in fact not the key concept concerning the representation of H, and that instead the input–output properties of H are determined, in general, by a certain type of family of responses. Some related material concerning other results, engineering education, and discrete‐space systems, is also given. Copyright © 2007 John Wiley & Sons, Ltd.

Let M be a $2\times 2$ matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1 , u2 , v1 , v2 with real coefficients and symmetry such that $$ \left[ \begin{matrix} u_1(z) &v_1(z)\\ u_2(z) &v_2(z) \end{matrix}\right] \left[ \begin{matrix} u_1(1/z) &u_2(1/z)\\ v_1(1/z) &v_2(1/z)\end{matrix}\right]=M(z) \qquad \forall\; z\in \CC \bs \{0 \} $$ and [Su1 ](z)[Sv2 ](z)=[Su2 ](z)[Sv1 ](z), where [Sp](z)=p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1 , u2 , v1 , v2 . As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters.

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

ON NECESSARY AND SUFFICIENT CONDITIONS FOR DUALS OF WAVELET FRAMES

Stability of nonlinear elliptic systems with distributed parameters and variable boundary data

In the beginning of the paper a short survey of development of the maximum principle is considered. In conclusion a sufficient condition of optimality in the form of the regular synthesis is formulated. This sufficient condition was recently obtained by the author. Under that condition all marked trajectories are robust optimal in the sense of termination. The robust optimality is essential for engineering problems

Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set Λ = { 0 , r / N } + 2 ⁢ ℤ {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of ℝ {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system { ψ j , λ ⁢ ( x ) = ( 2 ⁢ N ) j / 2 ⁢ ψ ⁢ ( ( 2 ⁢ N ) j ⁢ x - λ ) , j ∈ ℤ , λ ∈ Λ } {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for L 2 ⁢ ( ℝ ) {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

Materials science is an applied science concerned with the relationship between the struc- ture and properties of materials. Frames have become the focus of active research field, both in the- ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi- resolution are introduced. A precise exist-ence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient con-dition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

Continuous wavelet frames on the sphere: The group-theoretic approach revisited

Dimension functions play a significant role in the study of wavelets and have been attracting many waveletters’ interest. In recent years, the study of wavelet dimension functions has seen great achievements, but the study of Parseval frame wavelet (PFW) dimension functions has not. Bownik, Rzeszotnik and Speegle in 2001 and Arambašić, Bakić and Rajić in 2007 characterized ‐periodic functions that are wavelet dimension functions. But it is open what a ‐periodic function is qualified to be a dimension function of some semi‐orthogonal PFW. This paper addresses semi‐orthogonal PFW dimension functions associated with expansive matrices of determinant ± 2. We obtain a description of the ranges of semi‐orthogonal PFW dimension functions and establish a necessary and sufficient condition for an integer‐valued function to be a dimension function of some semi‐orthogonal PFW. Copyright © 2014 John Wiley & Sons, Ltd.

We study properties of the closure of the set of tight frame wavelets. We give a necessary condition and a sufficient condition for a function to be in this closure. In particular, we show that the collection of tight frame wavelets is not dense in L2(ℝn), which answers a question posed by D. Han and D. Larson (Preprint, 2008).

Métastases et pseudo-tumeurs pulmonaires: Metastasis and pulmonary pseudo-tumors