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1.
Gilbert Strang 《Physica D: Nonlinear Phenomena》1992,60(1-4):239-244
The construction of wavelets begins with a dilation equation (x) = Σck(2x-k). The choice of the coefficients ck controls the accuracy and orthogonality of wavelet expansions. We show how the Daubechies coefficients, which give pth order accuracy with 2p coefficients (the minimum number), follow from orthogonality and accuracy conditions applied to the polynomial Σckeikξ. 相似文献
2.
Jianhong Shen Gilbert Strang 《Proceedings of the American Mathematical Society》1996,124(12):3819-3833
To study wavelets and filter banks of high order, we begin with the zeros of . This is the binomial series for , truncated after terms. Its zeros give the zeros of the Daubechies filter inside the unit circle, by . The filter has additional zeros at , and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets with vanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros of . We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot for . The zeros approach a limiting curve in the complex plane, which is the circle . All zeros have , and the rightmost zeros approach (corresponding to ) with speed . The curve gives a very accurate approximation for finite . The wide dynamic range in the coefficients of makes the zeros difficult to compute for large . Rescaling by allows us to reach by standard codes.
3.
Approximation by translates of refinable functions 总被引:23,自引:0,他引:23
Summary.
The functions
are
refinable if they are
combinations of the rescaled and translated functions
.
This is very common in scientific computing on a regular mesh.
The space of approximating functions with meshwidth
is a
subspace of with meshwidth
.
These refinable spaces have refinable basis functions.
The accuracy of the computations
depends on , the
order of approximation, which is determined by the degree of
polynomials
that lie in .
Most refinable functions (such as scaling functions in the theory
of wavelets) have no simple formulas.
The functions
are known only through the coefficients
in the refinement equation – scalars in the traditional case,
matrices for multiwavelets.
The scalar "sum rules" that determine
are well known.
We find the conditions on the matrices
that
yield approximation of order
from .
These are equivalent to the Strang–Fix conditions on the Fourier
transforms
, but for refinable
functions they can be explicitly verified from
the .
Received
August 31, 1994 / Revised version received May 2, 1995 相似文献
4.
Maximal flow through a domain 总被引:1,自引:0,他引:1
Gilbert Strang 《Mathematical Programming》1983,27(2):123-143
It is proved that, if the DFP or BFGS algorithm with step-lengths of one is applied to a functionF(x) that has a Lipschitz continuous second derivative, and if the calculated vectors of variables converge to a point at which ?F is zero and ?2 F is positive definite, then the sequence of variable metric matrices also converges. The limit of this sequence is identified in the case whenF(x) is a strictly convex quadratic function. 相似文献
5.
6.
Approximation in the finite element method 总被引:2,自引:0,他引:2
Gilbert Strang 《Numerische Mathematik》1972,19(1):81-98
Summary The rate of convergence of the finite element method depends on the order to which the solutionu can be approximated by the trial space of piecewise polynomials. We attempt to unify the many published estimates, by proving that if the trial space is complete through polynomials of degreek–1, then it contains a functionv
h
such that |u–v
h
|
s
ch
k–s|u|
k
. The derivatives of orders andk are measured either in the maximum norm or in the mean-square norm, and the estimate can be made local: the error in a given element depends on the diameterh
i
of that element. The proof applies to domains in any number of dimensions, and employs a uniformity assumption which avoids degenerate element shapes.This research was supported by the National Science Foundation (GP-13778). 相似文献
7.
8.
Trees with Cantor Eigenvalue Distribution 总被引:2,自引:0,他引:2
We study a family of trees with degree k at all interior nodes and degree 1 at boundary nodes. The eigenvalues of the adjacency matrix have high multiplicities. As the trees grow, the graphs of those eigenvalues approach a piecewise-constant "Cantor function." For each value of , we will find the fraction of the eigenvalues that are given by . 相似文献
9.
Many authors have recognized that traffic under the traditional car‐following model (CFM) is subject to flow instabilities. A recent model achieves stability using bilateral control (BCM)—by looking both forward and backward [1]. (Looking back may be difficult or distracting for human drivers, but is not a problem for sensors.) We analyze the underlying systems of differential equations by studying their eigenvalues and eigenvectors under various boundary conditions. Simulations further confirm that bilateral control can avoid instabilities and reduce the chance of collisions. 相似文献
10.
Pavel Grinfeld Gilbert Strang 《Journal of Mathematical Analysis and Applications》2012,385(1):135-149
For regular polygons inscribed in a circle, the eigenvalues of the Laplacian converge as to the known eigenvalues on a circle. We compute the leading terms of in a series in powers of , by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The term comes from Hadamard?s formula, and reflects the change in area. This term disappears if we “transcribe” the polygon, scaling it to have the same area as the circle. 相似文献