共查询到20条相似文献,搜索用时 31 毫秒
1.
Guido Walz 《BIT Numerical Mathematics》1997,37(1):189-201
In this paper we investigate some properties of trigonometric B-splines. We establish a complex integral representation for
these functions, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different
and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a
trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. Finally we show that—in the case of equidistant
knots—the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve
possesses the convex-hull property. This is illustrated by a numerical example. 相似文献
2.
We investigate biorthogonal Wilson bases generated by cardinal B-splines. Such bases provide very efficient representations of sinusoids modulated by spline functions. Exact Riesz bounds for such bases are given. Furthermore, we derive an explicit formula for the computation of the dual window functions and determine their decay rates. 相似文献
3.
Daniela Roşca 《Results in Mathematics》2006,49(1-2):171-184
We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b. 相似文献
4.
In this paper we study B-splines when the intervals between consecutive knots are in geometric progression and obtain generalizations of the particularly simple properties of the uniform B-splines, where the knots are equally spaced. 相似文献
5.
The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL
2(R
d
) based on a general class of functions which includes polyharmonic B-splines.The work of this author has been partially supported by a DARPA grant.The work of this author has been partially supported by Fondo Nacional de Ciencia y Technologia under Grant 880/89. 相似文献
6.
This paper studies systems of tensor-product functions for which the functions they span are monotonic in any direction when
their control nets are monotonic in that direction. It is shown that Bernstein polynomials and B-splines have this property
but that totally positive systems in general, such as certain trigonometric and rational bases, do not.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
7.
Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles
with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect
to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis.
In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline
functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline
functions can be applied. 相似文献
8.
Gabor frames, unimodularity, and window decay 总被引:4,自引:0,他引:4
We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with
particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties
of g are inherited by its minimal dual γ0 and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying
(i.e., decay like |g(t)|≤Ce−α|t|
1/α for some 1/2≤α<1) window functions. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e.,
its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent
to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically
associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent
the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi
representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on
results from the theory of entire functions, and on the theory of polynomial matrices. 相似文献
9.
K. Mørken 《Constructive Approximation》1991,7(1):195-208
In this paper it is shown how the algebraic product of two spline functions, each represented in terms of B-splines, can again be represented as a linear combination of suitable B-splines. As a corollary to this result we obtain an explicit representation of a given B-spline function in terms of B-splines of some arbitrary higher degree. This generalizes some known results for raising the degree by one. Recurrence relations for both products and degree raising are established that may be useful for computation.Communicated by Larry L. Schumaker. 相似文献
10.
Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases. In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases. 相似文献
11.
Robert S. Strichartz 《Journal of Fourier Analysis and Applications》1997,3(4):387-416
Let SG denote the Sierpinski gasket with Hausdorff measure μ of dimensionlog 3/log 2, let PLk denote the continuous piecewise linear functions with respect to the usual triangulation of SG into 3k triangles, and let Wk denote the orthogonal complement of PLk−1 in PLk. We construct a basis for each Wk, so that the entire collection is a frame for L2(dμ). This wavelet basis is obtained from three wavelet generators by scaling, translation and rotation, and the wavelets
are supported either by one corner triangle or a pair of adjacent triangles in the triangulation of level k − 1. Analogous
bases are constructed in the von Koch curve, the hexagasket, and the n-dimensional analog of SG. 相似文献
12.
The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established. 相似文献
13.
J.W. Robbin 《Linear algebra and its applications》1975,10(2):95-102
Given a norm on a finite dimensional vector space V, we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory. 相似文献
14.
Thomas A. Hogan 《Journal of Fourier Analysis and Applications》1997,3(6):757-774
Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately,
several desirable properties are not available with compactly supported orthogonal wavelets, e.g., symmetry and piecewise
polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves
the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this article, we characterize
stability and linear independence of the shifts of a finite refinable function set in terms of the refinement mask. Several
illustrative examples are provided.
The characterizations given in Section 2 actually require that the refinable functions be minimal in some sense. This notion
of minimality is made clear in Section 3, where we provide sufficient conditions on the mask to ensure minimality. The conditions
are shown to be necessary also under further assumptions on the refinement mask. An example is provided illustrating how the
software package MAPLE can be used to investigate at least the case of two simultaneously refinable functions. 相似文献
15.
Hermite interpolation with trigonometric polynomials 总被引:10,自引:0,他引:10
Franz-Jürgen Delvos 《BIT Numerical Mathematics》1993,33(1):113-123
Interpolation methods of Hermite type in translation invariant spaces of trigonometric polynomials for any position of interpolation points and any number of derivatives are constructed. For the case of an odd number of interpolation conditions-periodic trigonometric polynomials of minimum order are chosen as interpolation functions while for the case of an even number of interpolation conditions-antiperiodic trigonometric polynomials of minimum order are appropriate. 相似文献
16.
Jin Liang Gaston M. N’Guérékata Ti-Jun Xiao Jun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2009
This paper is concerned with some properties of pseudo-almost automorphic functions, which are more general and complicated than pseudo-almost periodic functions. Using these properties, we establish an existence and uniqueness theorem for pseudo-almost automorphic mild solutions to semilinear differential equations in a Banach space. 相似文献
17.
A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding
integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and
multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported
refinable functions and refinable function vectors. 相似文献
18.
Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations
where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of
the transition operator associated with h.
Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999 相似文献
19.
Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions 总被引:7,自引:0,他引:7
This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions.
It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the
trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function.
Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be
biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function
for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with
desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory
orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric)
wavelets generated by interpolatory refinable functions.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
Kai Bittner 《Journal of Fourier Analysis and Applications》2002,8(1):85-108
Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only. 相似文献