A Gibbs phenomenon for spline functions

Richards, F. B.

doi:10.1016/0021-9045(91)90034-8

Cited by 45 publications

(55 citation statements)

References 6 publications

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“…This agrees with work on periodic spline approximations done by Richards [7]. He examined higher order splines in approximating the function…”

Section: A General Formula For Dyadic Sumssupporting

confidence: 69%

Gibbs Phenomenon for Wavelets

Kelly

1996

Applied and Computational Harmonic Analysis

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When a Fourier series is used to approximate a function with a jump discontinuity, an overshoot at the discontinuity occurs. This phenomenon was noticed by Michelson [6] and explained by Gibbs [3] in 1899. This phenomenon is known as the Gibbs effect. In this paper, possible Gibbs effects will be looked at for wavelet expansions of functions at points with jump discontinuities. Certain conditions on the size of the wavelet kernel will be examined to determine if a Gibbs effect occurs and what magnitude it is. An if and only if condition for the existence of a Gibbs effect is presented, and this condition is used to prove existence of Gibbs effects for some compactly supported wavelets. Since wavelets are not translation invariant, effects of a discontinuity will depend on its location. Also, computer estimates on the sizes of the overshoots and undershoots were computed for some compactly supported wavelets with small support. c 1996 Academic Press, Inc. GIBBS PHENOMENON FOR FOURIER SERIESTo illustrate what is happening in the Gibbs effect, let us examine the partial sums of a Fourier series. Let g(x) be a periodic, piecewise smooth function with a jump discontinuity at x 0 . For any fixed x 1 , not equal to x 0 , the partial sums of g(x) at x 1 approach g(x 1 ). That is, if s n is the partial sum of g, thenHowever, if x is allowed to approach the discontinuity as the partial sums are taken to the limit, an overshoot, or undershoot, may occur. That is,and1 E-mail: kelly@math.uwlax.edu.are possible. This overshoot, or undershoot, is called the Gibbs phenomenon.Proposition 1.1 Let f be a function of bounded variation, 2π-periodic function. At each jump discontinuity x 0 of f, the Fourier series for f will overshoot (undershoot). The overshoot and undershoot will be approximately 9% of the magnitude of the jump |f(xFor further details for the Fourier series, see [8]. GENERAL WAVELET STRUCTURE AND COMPACTLY SUPPORTED WAVELETSA general structure, called a multiresolution analysis, for wavelet bases in L 2 (R) was described by Mallat [5]. Letand there is a φ ∈ V 0 such that {φ m,n } n∈Z is an orthonormal basis of V m , whereDefine W m such that V m+1 = V m W m . Thus, L 2 (R) = W m . Then there exists a ψ ∈ W 0 such that {ψ m,n } n∈Z is an orthonormal basis of W m , and {ψ m,n } m,n∈Z is a wavelet basis of L 2 (R), where ψ m,n (x) = 2 m/2 ψ(2 m x − n).The function φ is called the scaling function, and ψ is called the mother function.

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“…This agrees with work on periodic spline approximations done by Richards [7]. He examined higher order splines in approximating the function…”

Section: A General Formula For Dyadic Sumssupporting

confidence: 69%

Gibbs Phenomenon for Wavelets

Kelly

1996

Applied and Computational Harmonic Analysis

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“…[k] (x)>1, then there exists a Gibbs phenomenon in the spline approximation of F. In [14] the Gibbs splines S…”

Section: [K]mentioning

confidence: 99%

“…Richards [14] studied the Gibbs phenomenon for the expansion into spline functions without directly referring to multiresolution theory. Let T…”

Section: Introductionmentioning

confidence: 99%

On the Gibbs Phenomenon for Wavelet Expansions

Shim

Volkmer

1996

Journal of Approximation Theory

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“…By increasing the order of approximation one can reduce the ripples, but cannot get rid of them completely. Richards [5] shows that the overshot tends to the typical 8.95% of shock magnitude as the degree of spline approximation approaches infinity. Almost all existing treatments for the Gibbs phenomenon reduction fall in the direction of summability (or averaging) methods, such as of Fejér [6] or Lanczos [7].…”

Section: Suppression Of the Gibbs Phenomenonmentioning

confidence: 97%

“…In this case each of the polynomials 'hops' over nodes of the partner polynomial. Figure 4 illustrates approximation of a step function by two fifth-degree hopping polynomials (HOPs), P (5) 1 at stencil {x 0 , x 2 , . .…”

Section: Suppression Of the Gibbs Phenomenonmentioning

confidence: 99%

Hopping numerical approximations of the hyperbolic equation

Tkalich

2007

Numerical Methods in Fluids

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SUMMARYPolynomial functions can be used to derive numerical schemes for an approximate solution of hyperbolic equations. A conventional derivation technique requires a polynomial to pass through every node values of a continuous computational stencil, leading to severe manifestation of the Gibbs phenomenon and strict time-step limitation. To overcome the problem, this paper introduces polynomials that skip regularly ('hop' over) one or more nodes from the computational grid. Polynomials hopping over odd and even nodes yield a series of explicit numerical schemes of a required accuracy, with Lax-Friedrichs method being a particular simplest case. The schemes have two times wider stability interval compared to conventional continuous-stencil explicit methods. Convex combinations of odd-and even-node-based updates improve further accuracy and stability of the method. Out of considered combinations (up to third-order accuracy), derived odd-order methods are stable for the Courant number ranging from 0 to 3, and even-order ones from 0 to 5. A 2-D extension of the hopping polynomial method exhibits similar properties.

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A Gibbs phenomenon for spline functions

Cited by 45 publications

References 6 publications

Gibbs Phenomenon for Wavelets

Gibbs Phenomenon for Wavelets

On the Gibbs Phenomenon for Wavelet Expansions

Hopping numerical approximations of the hyperbolic equation

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