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23
Nonstationary Wavelets on the mSphere for Scattered Data
, 1996
"... We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the msphere. The wavelets are intrinsically defined on the msphere, and are independent of the choice of coordi ..."
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Cited by 47 (6 self)
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We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the msphere. The wavelets are intrinsically defined on the msphere, and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2sphere, we derive an uncertainty principle that expresses the tradeoff between localization and the presence of high harmonicsor high frequenciesin expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct. I. Introduction Geophyiscal or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. Synthesizing and analyzing such data is the motivation for the work that is pr...
Wavelets Associated with Periodic Basis Functions
"... In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decompositio ..."
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Cited by 35 (4 self)
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In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decomposition and reconstruction coefficients can be computed in terms of the discrete Fourier transform, so that FFT methods apply for their evaluation. In addition, decomposition at the n th level only involves 2 terms from the higher level. Similar remarks apply for reconstruction. We apply a periodic "uncertainty principle" to obtain an angle/frequency uncertainty "window" for these wavelets, and we show that for many wavelets in this class the angle/frequency localization is good.
On the Detection of Singularities of a Periodic Function
 ADV. COMPUT. MATH
"... We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometri ..."
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Cited by 28 (15 self)
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We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series.
Interpolatory And Orthonormal Trigonometric Wavelets
 J. Zeevi and R. Coifman
, 1998
"... The aim of this paper is the detailed investigation of trigonometric polynomial spaces as a tool for approximation and signal analysis. Sample spaces are generated by equidistant translates of certain de la Vallée Poussin means. The different de la Vallée Poussin means enable us to choose between be ..."
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Cited by 24 (13 self)
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The aim of this paper is the detailed investigation of trigonometric polynomial spaces as a tool for approximation and signal analysis. Sample spaces are generated by equidistant translates of certain de la Vallée Poussin means. The different de la Vallée Poussin means enable us to choose between better time or frequencylocalization. For nested sample spaces and corresponding wavelet spaces, we discuss different bases and their transformations.
Polynomial Wavelets on the Interval
 Constr. Approx
, 1994
"... We investigate a polynomial wavelet decomposition of the L²(1, 1)space with Chebyshevweight, where the wavelets fulfill certain interpolatory conditions. For this approach we obtain the twoscale relations and decomposition formulas. Dual functions and Rieszstability will be discussed. ..."
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Cited by 15 (3 self)
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We investigate a polynomial wavelet decomposition of the L&sup2;(1, 1)space with Chebyshevweight, where the wavelets fulfill certain interpolatory conditions. For this approach we obtain the twoscale relations and decomposition formulas. Dual functions and Rieszstability will be discussed.
On local smoothness classes of periodic functions
 J. Fourier Anal. Appl
"... We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of d ..."
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Cited by 10 (7 self)
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We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order. 1
Trigonometric Wavelets and the Uncertainty Principle
 Math. Research
, 1995
"... The timefrequency localization of trigonometric wavelets is discussed. A good measure is provided by a periodic version of the Heisenberg uncertainty principle. We consider multiresolution analyses generated by de la Vall'ee Poussin means of the Dirichlet kernel. For the resulting interpolator ..."
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Cited by 9 (3 self)
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The timefrequency localization of trigonometric wavelets is discussed. A good measure is provided by a periodic version of the Heisenberg uncertainty principle. We consider multiresolution analyses generated by de la Vall'ee Poussin means of the Dirichlet kernel. For the resulting interpolatory and orthonormal scaling functions and wavelets, the uncertainty product can be bounded from above by a constant. 1 Introduction In the Hilbert space L 2 (R), the wellknown Heisenberg uncertainty principle can be phrased as \Deltat(f ) \Delta!(f ) ß 2 kfk 4 ; where, for f 2 L 2 2ß , \Deltat(f ) := 1 Z \Gamma1 (t \Gamma t 0 ) 2 jf(t)j 2 dt ; t 0 := 1 Z \Gamma1 t jf(t)j 2 dt ; \Delta!(f ) := 1 Z \Gamma1 (! \Gamma ! 0 ) 2 j f(!)j 2 d! ; ! 0 := 1 Z \Gamma1 ! j f(!)j 2 d! : As Meyer has pointed out [3, 4], basis functions with good timefrequency localization behaviour can be thought of as functions f h = h \Gamma1=2 f(\Delta=h); h ? 0; satisfying \Delt...
Polynomial Frames for the Detection of Singularities
"... We propose a class of algebraic polynomial frames, which are computationally easier to implement than polynomial bases. We also discuss the weighted L p  stability of our frames for 1 # p # #. Our analysis is based on orthogonal polynomials with respect to the weight in question, but the fram ..."
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Cited by 7 (3 self)
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We propose a class of algebraic polynomial frames, which are computationally easier to implement than polynomial bases. We also discuss the weighted L p  stability of our frames for 1 # p # #. Our analysis is based on orthogonal polynomials with respect to the weight in question, but the frame bounds are independent of the system of orthogonal polynomials used. In spite of the fact that algebraic polynomials are inherently nonlocal, our frames provide good localization properties. In particular, they can be used to detect discontinuities in derivatives of all orders of a function. We describe asymptotic expressions for the frame coefficients in the vicinity of a discontinuity.
Interpolatory Wavelets on the Sphere
 IN APPROXIMATION THEORY VIII
, 1995
"... In this paper, we construct an interpolatory wavelet basis on the unit sphere S ae R³. Using spherical coordinates, we apply the tensor product of interpolatory trigonometric and algebraic polynomial wavelets. The described decomposition and reconstruction algorithms work in the frequency domain. ..."
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Cited by 6 (1 self)
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In this paper, we construct an interpolatory wavelet basis on the unit sphere S ae R³. Using spherical coordinates, we apply the tensor product of interpolatory trigonometric and algebraic polynomial wavelets. The described decomposition and reconstruction algorithms work in the frequency domain.
Polynomial frames: a fast tour
 In Approximation theory XI: Gatlinburg 2004, Mod. Methods Math. Nashboro
, 2005
"... Abstract. We present a unifying theme in an abstract setting for some of the recent work on polynomial frames on the circle, the unit interval, the real line, and the Euclidean sphere. In particular, we describe a construction of a tight frame in the abstract setting, so that certain Besov approxima ..."
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Cited by 5 (2 self)
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Abstract. We present a unifying theme in an abstract setting for some of the recent work on polynomial frames on the circle, the unit interval, the real line, and the Euclidean sphere. In particular, we describe a construction of a tight frame in the abstract setting, so that certain Besov approximation spaces can be characterized using the absolute values of the frame coefficients. We discuss the localization properties of the frames in the context of trigonometric, Jacobi, and spherical polynomials, and discuss some applications. Wavelet analysis is perhaps one of the fastest growing areas of analysis. Some of the applications include image processing, signal processing, time series analysis, probability density estimation, neural networks, data compression, etc. Traditionally [4, 6, 39], wavelets are defined using the notion