Communications on Pure and Applied Mathematics
0010-3640
1097-0312
Mỹ
Cơ quản chủ quản: WILEY , Wiley-Liss Inc.
Lĩnh vực:
Mathematics (miscellaneous)Applied Mathematics
Các bài báo tiêu biểu
Orthonormal bases of compactly supported wavelets Abstract We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.
Tập 41 Số 7 - Trang 909-996 - 1988
Stable signal recovery from incomplete and inaccurate measurements Abstract Suppose we wish to recover a vector x 0 ∈ ℝ𝓂 (e.g., a digital signal or image) from incomplete and contaminated observations y = A x 0 + e ; A is an 𝓃 × 𝓂 matrix with far fewer rows than columns (𝓃 ≪ 𝓂) and e is an error term. Is it possible to recover x 0 accurately based on the data y ? To recover x 0 , we consider the solution x # to the 𝓁1 ‐regularization problem
where ϵ is the size of the error term e . We show that if A obeys a uniform uncertainty principle (with unit‐normed columns) and if the vector x 0 is sufficiently sparse, then the solution is within the noise level
As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A 's provided that the number of nonzeros of x 0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x 0 ; then stable recovery occurs for almost any set of 𝓃 coefficients provided that the number of nonzeros is of the order of 𝓃/(log 𝓂)6 . In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals. © 2006 Wiley Periodicals, Inc.
Tập 59 Số 8 - Trang 1207-1223 - 2006
Optimal approximations by piecewise smooth functions and associated variational problems
Tập 42 Số 5 - Trang 577-685 - 1989
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Abstract We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted 𝓁p p ≤ 2, still regularizes the problem. Use of such 𝓁p p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.
Tập 57 Số 11 - Trang 1413-1457 - 2004
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
Tập 12 Số 4 - Trang 623-727 - 1959
Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
Tập 36 Số 4 - Trang 437-477 - 1983
Biorthogonal bases of compactly supported wavelets Abstract Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.
Tập 45 Số 5 - Trang 485-560 - 1992
On the exponential solution of differential equations for a linear operator
Tập 7 Số 4 - Trang 649-673 - 1954