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Nonlinear Approximation
 AMER. J. MATH
, 2006
"... We prove that if X is any 2regular scheme (in the sense of CastelnuovoMumford) then X is small. This means that if L is a linear space and Y := L X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y 1. The converse is true and wellk ..."
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We prove that if X is any 2regular scheme (in the sense of CastelnuovoMumford) then X is small. This means that if L is a linear space and Y := L X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y 1. The converse is true and wellknown for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2regular", is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X is 2regular, then so is X red , and so also is the projection of X from any point of X. Our result
Adaptive Nonlinear Approximations
, 1994
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best matc ..."
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Cited by 73 (1 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best
Nonlinear Approximation with Walsh Atoms
 Surface Fitting and Multiresolution Methods
, 1997
"... . As a model for nonlinear approximation from a redundant set of timefrequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations of pai ..."
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Cited by 13 (3 self)
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. As a model for nonlinear approximation from a redundant set of timefrequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations
Nonlinear approximation by trigonometric sums
 J. Fourier Anal. and Appl
, 1995
"... ABSTRACT. We investigate the Lperror of approximation to a function f ∈ Lp(Td) by a linear combination ∑ k ckek of n exponentials ek(x): = ei〈k,x 〉 = ei(k1x1+···+kd xd) d on T, where the frequencies k ∈ Zd are allowed to depend on f. We bound this error in terms of the smoothness and other propert ..."
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Cited by 23 (7 self)
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ABSTRACT. We investigate the Lperror of approximation to a function f ∈ Lp(Td) by a linear combination ∑ k ckek of n exponentials ek(x): = ei〈k,x 〉 = ei(k1x1+···+kd xd) d on T, where the frequencies k ∈ Zd are allowed to depend on f. We bound this error in terms of the smoothness and other
Approximate Signal Processing
, 1997
"... It is increasingly important to structure signal processing algorithms and systems to allow for trading off between the accuracy of results and the utilization of resources in their implementation. In any particular context, there are typically a variety of heuristic approaches to managing these tra ..."
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Cited by 516 (2 self)
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these tradeoffs. One of the objectives of this paper is to suggest that there is the potential for developing a more formal approach, including utilizing current research in Computer Science on Approximate Processing and one of its central concepts, Incremental Refinement. Toward this end, we first summarize a
On Nonuniqueness in Nonlinear ...Approximation
"... : It is shown that under weak assumptions nonlinear L 2 approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential family of approximating functions. In addition, for a certain class of approximating families, we construct f ..."
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: It is shown that under weak assumptions nonlinear L 2 approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential family of approximating functions. In addition, for a certain class of approximating families, we construct
ASPECTS OF NONLINEAR APPROXIMATION WITH DICTIONARIES by
"... 5Denne afhandling er af de Ingeniør, Natur og Sundhedsvidenskabelige Fakulteter ved Aalborg Universitet antaget til offentligt forsvar for den naturvidenskabelige doktorgrad. Forsvaret finder sted i NOVI Auditoriet, Niels Jernes Vej 10, den 13. marts 2009 kl. 13.00 præcis. Aalborg den 9. januar 2 ..."
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5Denne afhandling er af de Ingeniør, Natur og Sundhedsvidenskabelige Fakulteter ved Aalborg Universitet antaget til offentligt forsvar for den naturvidenskabelige doktorgrad. Forsvaret finder sted i NOVI Auditoriet, Niels Jernes Vej 10, den 13. marts 2009 kl. 13.00 præcis. Aalborg den 9. januar 2009
NONLINEAR APPROXIMATION IN αMODULATION SPACES
, 2003
"... ABSTRACT. The αmodulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the αmodulation spaces. We study mterm nonlinear approximation with ..."
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Cited by 10 (3 self)
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ABSTRACT. The αmodulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the αmodulation spaces. We study mterm nonlinear approximation
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