• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.759-769
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• 2005

We first discuss jump discontinuity in higher dimension, and then prove a local convergence theorem for wavelet approximations in higher dimension. We also redefine the concept of Gibbs phenomenon in higher dimension and show that wavelet expansion exhibits Gibbs phenomenon.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.605-612
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• 2004

The Gibbs’ phenomenon for the classical Fourier series is known. This occurs for almost all series expansions. This phenomenon has been observed even in sampling series. In this paper, we show the existence of Gibbs phenomenon for trigonometric interpolating polynomial by a simple and different manner from the wok[4].

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.719-736
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• 2005

An understanding of Fourier series and their generalization is important for physics and engineering students, as much for mathematical and physical insight as for applications. Students are usually confused by the so-called Gibbs' phenomenon, an overshoot between a discontinuous function and its approximation by a Fourier series as the number of terms in the series becomes indefinitely large. In this paper we give short story of Gibbs phenomenon in chronological order.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.89-98
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• 2011

The Fourier series has a rapid oscillation near end points at jump discontinuity which is called the Gibbs phenomenon. There is an overshoot (or undershoot) of approximately 9% at jump discontinuity. In this paper, we prove that a bunch of series representations (certain nonharmonic Fourier series) give good approximations vanishing Gibbs phenomenon. Also we have an application for approximating some shape of upper part of a vehicle in a different way from the method of cubic splines and wavelets.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.507-517
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• 2000

Existence of Gibbs' phenomenon has been well known in wavelet expansions. But the estimation of its size is another problem. Because of the oscillation of wavelets, it is not easy to estimate the Gibbs size of wavelet expansions. For wavelets defined via Fourier transforms, we give a new formula to calculate the size of overshoot. But using this we compute the size of Gibbs effect for Barttle-Lemarier wavelets.

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.181-193
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• 1998

The Shannon sampling series is the prototype of an interpolating series or sampling series. Also the Shannon wavelet is one of the protypes of wavelets. But the coefficients of the Shannon sampling series are different function values at the point of discontinuity, we analyze the Gibbs phenomenon for the Shannon sampling series. We also find a way to go around this overshoot effect.

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.199-209
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• 2003

When a sampling theorem holds in wavelet subspaces, sampling expansions can be a good approximation to projection expansions. Even when the sampling theorem does not hold, the scaling function series with the usual coefficients replaced by sampled function values may also be a good approximation to the projection. We refer to such series as hybrid sampling series. For this series, we shall investigate the local convergence and analyze Gibbs phenomenon.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.613-620
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• 2004

The concept of Gibbs’ phenomenon has not been made for higher dimension in wavelets. In this paper we extend the concept in two dimensional wavelets. We give the fundamental concept of jump discontinuity in two dimensions. We provide the criteria for the existence of Gibbs phenomenon for both separable and tensor product wavelets.

• Journal of applied mathematics & informatics
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• v.9 no.2
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• pp.657-666
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• 2002

ThE Gibbs' phenomenon in the classical Fourier series is well-known. It is closely related with the kernel of the partial sum of the series. In fact, the Dirichlet kernel of the courier series is not positive. The poisson kernel of Cesaro summability is positive. As the consequence of the positiveness, the partial sum of Cesaro summability does not exhibit the Gibbs' phenomenon. Most kernels associated with wavelet expansions are not positive. So wavelet series is not free from the Gibbs' phenomenon. Because of the excessive oscillation of wavelets, we can not follow the techniques of the courier series to get rid of the unwanted quirk. Here we make a positive kernel For Meyer wavelets and as the result the associated summability method does not exhibit Gibbs' phenomenon for the corresponding series .

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.633-641
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• 2004

In this paper we deal with functions in higher dimension. We provide several convergence theorem for approximation by convolution type delta sequence. We also give sufficient and necessary condition for Gibbs phenomenon to exist.