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Orthogonal polynomials for refinable linear functionals [PDF]
Mathematics of Computation, 2006 A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function.
Dirk Laurie, Johan de Villiers 0001
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Structure of refinable splines [PDF]
, 2007 A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs.
Wang, Yang +4 more
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Uniformly refinable maps [PDF]
, 2023 [EN] We introduce the notion of uniformly refinable map for compact, Hausdorff spaces, as a generalization of refinable maps originallydefined for metric continua by Jo Ford (Heath) and Jack W. Rogers, Jr., Refinable maps, Colloq.
Macías, Sergio
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Neural Network Approximation of Refinable Functions
IEEE Transactions on Information Theory, 2023 In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising ...
Ingrid Daubechies +8 more
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Symmetric multivariate orthogonal refinable functions [PDF]
, 2004 In this paper, we shall investigate the symmetry property of a multivariate orthogonal M-refinable function with a general dilation matrix M. For an orthogonal M-refinable function ϕ such that ϕ is symmetric about a point (centro-symmetric) and ϕ ...
Han, Bin
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The refinability of step functions [PDF]
Proceedings of the American Mathematical Society, 2007 Refinable functions have been widely investigated because of their importance in wavelet theory and multiresolution analysis, as well as because of intrinsic interest. Problems involving refinability can be challenging and interesting problems in mathematics. Several papers have investigated refinability of splines and other classes of functions.
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, 2007 A refinable set is a compact set with positive Lebesgue measure whose characteristic function satisfies a refinement equation. Refinable sets are a generalization of self-affine tiles.
Wang, Yang, Dai, Xinrong
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Multivariate Refinable Interpolating Functions
Applied and Computational Harmonic Analysis, 1999 The author gives an algorithm for the construction of refinable interpolating functions for an arbitrary dilation matrix. This construction of refinable interpolating functions is an intermediate step in the construction of orthonormal wavelet bases and is of interest in its own right.
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Classification of refinable splines
, 2006 A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design.
Wang, Yang, Dai, Xin Rong, Feng, De Jun
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Gauss Quadrature for Refinable Weight Functions
Applied and Computational Harmonic Analysis, 2000 A refineable function \(\phi\) is a solution of a two-scale difference equation \[ \phi(x)= \sum_{j\in\mathbb{Z}} a_j\phi(2x- j), \] where \(a_j\) are real numbers satisfying \[ \sum_{j\in\mathbb{Z}} a_{k+ 2j}= 1,\quad\text{all }k\in\mathbb{Z}. \] The authors study Gaussian quadrature rules having refineable functions as weight functions.
Gautschi, Walter +2 more
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