scholarly journals On a class of splines free of Gibbs phenomenon

2020 ◽  
Author(s):  
Sergio Amat ◽  
Juan Ruiz ◽  
Chi-Wang Shu ◽  
Juan Carlos Trillo
Keyword(s):  
Spline Interpolation ◽  
Gibbs Phenomenon ◽  
Jump Discontinuity ◽  
Regularity Conditions ◽  
Weighted Mean ◽  
Piecewise Polynomials ◽  
Jump Discontinuities ◽  
Nonlinear Technique ◽  
The Right ◽  
Theoretical Accuracy

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.

scholarly journals Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Beong In Yun
Keyword(s):  
Target Function ◽  
Gibbs Phenomenon ◽  
Weighted Averaging ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Left Hand ◽  
Partial Sum ◽  
Hand Part ◽  
Approximate Function ◽  
The Right

We introduce a generalized sigmoidal transformation wm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(r;x), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.

scholarly journals A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 335
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Gibbs Phenomenon ◽  
Order Of Approximation ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Numerical Examples ◽  
Piecewise Polynomials ◽  
Jump Discontinuities ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Numerical Order

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials

2019 ◽  
Vol 63 (3) ◽  
pp. 655-669
Author(s):  
A. Pepin ◽  
S. S. Beauchemin ◽  
S. Léger ◽  
N. Beaudoin
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Spline Interpolation ◽  
Higher Order ◽  
New Method ◽  
System Of Equations ◽  
Piecewise Polynomials ◽  
Odd Degree ◽  
High Degree

AbstractEffective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.

Solvability of cardinal spline interpolation problems

1983 ◽  
Vol 95 (1-2) ◽  
pp. 39-57
Author(s):  
T. N. T. Goodman
Keyword(s):  
Incidence Matrix ◽  
Bounded Solution ◽  
Spline Interpolation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomials ◽  
Cardinal Spline ◽  
Interpolation Problems ◽  
Bounded Data ◽  
Derivatives Of ◽  
Cardinal Spline Interpolation

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

scholarly journals Gibbs phenomena for Lq-best approximation in finite element spaces

2021 ◽  
Author(s):  
Sarah Roggendorf ◽  
Paul Houston ◽  
Kristoffer van der Zee
Keyword(s):  
Finite Element ◽  
Sufficient Conditions ◽  
Gibbs Phenomenon ◽  
Two Dimensions ◽  
Thin Layers ◽  
Jump Discontinuities ◽  
Recent Developments ◽  
Minimum Residual ◽  
Gibbs Phenomena ◽  
Theoretical Results

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

SEMI-NONPARAMETRIC ESTIMATES OF CURRENCY SUBSTITUTION BETWEEN THE CANADIAN DOLLAR AND THE U.S. DOLLAR

2009 ◽  
Vol 14 (1) ◽  
pp. 29-55 ◽  
Author(s):  
Apostolos Serletis ◽  
Guohua Feng
Keyword(s):  
Exchange Rate ◽  
Foreign Exchange Market ◽  
Exchange Rate Regime ◽  
The United States ◽  
Microeconomic Theory ◽  
Regularity Conditions ◽  
Bank Of Canada ◽  
Nonparametric Estimates ◽  
The Right ◽  
Monetary Assets

In this paper we investigate the issue of whether a floating currency is the right exchange rate regime for Canada or whether Canada should consider a currency union with the United States. In the context of the framework recently proposed by James L. Swofford, we use a semi-nonparametric flexible functional form—the asymptotically ideal model (AIM), introduced by William A. Barnett and A. Jonas—and pay explicit attention to the theoretical regularity conditions of neoclassical microeconomic theory, following the suggestions of William A. Barnett and William A. Barnett and Meenakshi Pasupathy. Our results indicate that U.S. dollar deposits are complements to domestic (Canadian) monetary assets, suggesting that Canada should continue the current exchange rate regime, allowing the exchange rate to float freely with no intervention in the foreign exchange market by the Bank of Canada.

The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices

1962 ◽  
Vol 14 ◽  
pp. 723-728 ◽  
Author(s):  
W. T. Sledd
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Summability Method ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Lebesgue Constants ◽  
Dirichlet Conditions ◽  
Image Position

Ifψ(x)is a real-valued function which has a jump discontinuity atx= ε and otherwise satisfies the Dirichlet conditions in a neighbourhood ofx= ε then{sn(x)}the sequence of partial sums of the Fourier series forψ(x)cannot converge uniformly atx =ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such thattn→ ε andThis behaviour of{sn(x)}is called the Gibbs phenomenon. If {σn(x)} is the transform of{sn(x)}by a summability methodT, and if {σn(x)} also has the property described then we say thatTpreserves the Gibbs phenomenon.

scholarly journals Doppler Radar Observations of Anticyclonic Tornadoes in Cyclonically Rotating, Right-Moving Supercells

2016 ◽  
Vol 144 (4) ◽  
pp. 1591-1616 ◽  
Author(s):  
Howard B. Bluestein ◽  
Michael M. French ◽  
Jeffrey C. Snyder ◽  
Jana B. Houser
Keyword(s):  
Case Studies ◽  
Doppler Radar ◽  
Environmental Parameters ◽  
Radar Data ◽  
Doppler Velocity ◽  
Weighted Mean ◽  
Radar Observations ◽  
Azimuthal Shear ◽  
The Right ◽  
Gust Front

Abstract Supercells dominated by mesocyclones, which tend to propagate to the right of the tropospheric pressure-weighted mean wind, on rare occasions produce anticyclonic tornadoes at the trailing end of the rear-flank gust front. More frequently, mesoanticyclones are found at this location, most of which do not spawn any tornadoes. In this paper, four cases are discussed in which the formation of anticyclonic tornadoes was documented in the plains by mobile or fixed-site Doppler radars. These brief case studies include the analysis of Doppler radar data for tornadoes at the following dates and locations: 1) 24 April 2006, near El Reno, Oklahoma; 2) 23 May 2008, near Ellis, Kansas; 3) 18 March 2012, near Willow, Oklahoma; and 4) 31 May 2013, near El Reno, Oklahoma. Three of these tornadoes were also documented photographically. In all of these cases, a strong mesocyclone (i.e., vortex signature characterized by azimuthal shear in excess of ~5 × 10−3 s−1 or a 20 m s−1 change in Doppler velocity over 5 km) or tornado was observed ~10 km away from the anticyclonic tornado. In three of these cases, the evolution of the tornadic vortex signature in time and height is described. Other features common to all cases are noted and possible mechanisms for anticyclonic tornadogenesis are identified. In addition, a set of estimated environmental parameters for these and other similar cases are discussed.

The Gibbs Phenomenon for Generalized Taylor and Euler Transforms

1975 ◽  
Vol 27 (2) ◽  
pp. 384-395
Author(s):  
Robert E. Powell ◽  
Richard A. Shoop
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Dirichlet Conditions

Let f be a real-valued function satisfying the Dirichlet conditions in a neighborhood of x = x0, at which point f has a jump discontinuity. If {Sn(x)} is the sequence of partial sums of the Fourier series of f at x, then ﹛Sn(x)﹜ cannot converge uniformly at x — x0. Moreover, for any number , there exists a sequence ﹛tn﹜, where tn → x0 and

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