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Normed groups: dichotomy and duality
"... The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a ri ..."
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Cited by 21 (14 self)
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The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a rightinvariant metric carrying ‡ows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closedgraph theorem, to the Birkho¤Kakutani metrization theorem and further back still to Banach’s Théorie des opérations linéaires. We collect together known salient features and develop their theory including Steinhaus theory uni…ed by the Category Embedding Theorem [BOst11], the associated themes of subadditivity and convexity, and a topological duality inherent to topological
Ridge functions and orthonormal ridgelets
 J. Approx. Theory
"... Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coeff ..."
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Cited by 20 (2 self)
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Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills ’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1D wavelet coefficients of the 1D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best mterm approximation. Thresholding of wavelet coefficients.
Nonlinear approximation from differentiable piecewise polynomials
 SIAM J. Math. Anal
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Normed versus topological groups: dichotomy and duality
 DISSERTATIONES MATH
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Duality and interpolation of anisotropic Triebel–Lizorkin spaces
 MATHEMATISCHE ZEITSCHRIFT
, 2008
"... We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on Rn using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469– 1510, 2006), where we generalized the isot ..."
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Cited by 7 (1 self)
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We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on Rn using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469– 1510, 2006), where we generalized the isotropic methods of dyadic ϕtransforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to nonisotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces.
Nonlinear wavelet approximation in anisotropic Besov spaces
 Indiana University Mathematics Journal
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Interpolation functors and Banach couples
 Actes Congr`es Intern. Math. 1970
, 1971
"... The theory of interpolation spaces originally arose from an attempt to generalize the classical interpolation theorems of M. Riesz and Marcinkiewicz to a more abstract setting. However it should more correctly be described as a theory of "families " of abstract spaces: Given a number of ( ..."
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Cited by 1 (0 self)
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The theory of interpolation spaces originally arose from an attempt to generalize the classical interpolation theorems of M. Riesz and Marcinkiewicz to a more abstract setting. However it should more correctly be described as a theory of "families " of abstract spaces: Given a number of (usually two) spaces contained
Absolute and Relative CutOff in Adaptive Approximation By Wavelets
, 1996
"... Given the wavelet expansion of a function v, a nonlinear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sob ..."
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Cited by 1 (1 self)
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Given the wavelet expansion of a function v, a nonlinear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sobolev or Besov scale) in terms of summability of properly defined subsets of its coefficients. Apriori estimates of the approximation error are derived. For the Haar system, the asymptotic behavior of both the approximation error and the number of survived coefficients is thoroughly investigated for a class of functions having Höldertype singularities.
Spaces of Dominating Mixed Smoothness – the Case of Compact Embeddings
, 2010
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Best mTerm Approximation and SobolevBesov
Spaces
, 2009
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Best mterm Approximation and LizorkinTriebel