Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis

Abstract

ABSTRACT. We deal with the maximum Gibbs ripple of the sampling wavelet series of a discontinuous .function f at a point t ~ R, .for all possible values o.['a satisfying f (t) = ee.f (t -0) + (1 -cO.f (t + 0). For the Shannon wavelet series, we make a complete description of all ripples, .for any ot in [0,1]. We show that Meyer sampling series exhibit Gibbs Phenomenon.lor ce < 0.12495 and ct > 0.306853. We also give Meyer sampling formulas with maximum overshoots shorter than Shannon's for several et in [0,1].

On the Gibbs–Wilbraham Phenomenon for Sampling and Interpolatory Series

“…With these considerations in mind, we make the following evident definition (cf. [1]). • If f is a bounded function such that f (0−) < f(0+), then φ exhibits a left (respectively, right) Gibbs-Wilbraham phenomenon for f provided there exists a y < 0 (respectively, an…”

“…Since then, the phenomenon has been explored for many other approximation methods, including spline interpolation [7,17] and wavelet expansions [1,2,13,15,20]. For more examples and a survey of the literature, see [14].…”

“…Gibbs's observation was precisely that despite the fact that the partial sums of the Fourier series of a periodic function converge pointwise to the function (or to the average value 1 2 (f (t+) + f (t−)) across a jump discontinuity), the graph of the limit (i.e. of f ), is not the (visual) limit of the graphs (of S N [f ]).…”

New Bounds for Truncation-Type Errors on Regular Sampling Expansions

“…Given a closed subspace U of L 2 ðRÞ and f 2 U , h ¼ À1 and N 1 < N 2 ðN 1 , N 2 2 ZÞ, the function: is called the Truncation Error of the sampling expansion (2). Given a closed subspace U of L 2 ðRÞ and f 2 U , h ¼ À1 and N 1 < N 2 ðN 1 , N 2 2 ZÞ, the function: is called the Truncation Error of the sampling expansion (2).…”

Local Sampling Problems

“…The earliest results on local sampling may be found in [7,9]. In [2], N. Atreas et al investigated the local error of the reconstruction formula in a multiresolution analysis (MRA). In [14], S. Y. Yang find a new method to evaluate the asymptotic rate of decay of the sampling function, and generalized their result to higher dimensional cases.…”