&lt;title&gt;Biorthogonal Wilson bases&lt;/title&gt;

Bittner, Kai

doi:10.1117/12.366798

Cited by 10 publications

(13 citation statements)

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“…In the following, we will consider biorthogonal Wilson bases, which are described in [5,7,8]. Here, we recall briefly the main results.…”

Section: Biorthogonal Wilson Basesmentioning

confidence: 99%

“…Oversampled Wilson systems have been investigated in [10]. In [5,7,8], the author has given a theory of biorthogonal Wilson bases, where the window functions do not need to be symmetric and even two different window functions are possible. In particular, an explicit formula for the computation of Riesz bounds is given which uses the Zak transform of the window function.…”

Section: Introductionmentioning

confidence: 99%

“…[11,13]) instead of the mixture of 1-periodic cosines and sines D k j chosen in [14]. For each of these choices one obtains analogous basis and approximation properties (see [5,7,8]). Here, we will restrict ourselves to the case of anti-periodic cosines C k j .…”

Section: Introductionmentioning

confidence: 99%

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Linear Approximation and Reproduction of Polynomials by Wilson Bases

Bittner¹

2002

Journal of Fourier Analysis and Applications

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Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only.

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“…In the following, we will consider biorthogonal Wilson bases, which are described in [5,7,8]. Here, we recall briefly the main results.…”

Section: Biorthogonal Wilson Basesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear Approximation and Reproduction of Polynomials by Wilson Bases

Bittner¹

2002

Journal of Fourier Analysis and Applications

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“…Wilson bases without the two-overlapping restriction have been investigated by several authors [8,11,16,17,19]. In [12], window functions which reproduce linear functions with only the first frequency term were constructed, allowing the overlap of three windows at the same time.…”

Section: Introductionmentioning

confidence: 99%

Spline Modulation of Sinusoids for Signal Representation

Bittner

Chui

2003

Journal of Fourier Analysis and Applications

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“…For instance, studies that focus on its connection with Gabor frames (Bownik et al 2017;Kaushik and Panwar 2013), construction of Wilson bases with arbitrary shapes (Bittner 1999;Chui and Shi 2000), or studies of discretized Wilson bases (Bolcskei et al 1996;Kutyniok and Strohmer 2005). So far, the use of Wilson bases in electromagnetics has only been advocated by Arnold (2002).…”

Section: Introductionmentioning

confidence: 99%

Wilson basis expansions of electromagnetic wavefields: a suitable framework for fiber optics

Floris

Hon

2018

Opt Quant Electron

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To detach large vectorial full-wave electromagnetic scattering problems from the specific, dissimilar bases that are typically associated with the optical waveguiding structures comprising an optical interface, we propose and demonstrate the expansion of wavefields in a single universal Wilson basis. We construct a Wilson basis by following the derivation in the paper by Daubechies, Jaffard and Journé. This basis features exponentially decaying basis functions in both the spatial and spectral domain. The strong localization in phase-space, with basis functions that strongly resemble wavefields, allow for efficient expansions of high-frequency electromagnetic fields. In a Wilson basis, the interface scattering problem is effectively separated from the physical configuration. For the evaluation of multiple, laterally displaced interface configurations, one may reuse electromagnetic fields in the Wilson basis, because the translation operator is sparse and diagonally dominant. We consider actual reflection-transmission problems comprising optical fibers and homogeneous media in the Wilson basis framework in a companion paper. There, the localization in the spectral domain aids the conversion of the numerical scheme to generate electromagnetic fields in homogeneous media due to Wilson-basis discretized electromagnetic sources. In this paper, we review the Wilson basis construction, demonstrate the expansion of modal electromagnetic fields in an optical fiber, and complex-source beams that are tilted with respect to the optical axis. Even for largely tilted beams (up to 60°), despite being highly oscillatory in the cross-sectional plane, the fields are well represented by a finite number of higher-order Wilson basis functions.

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<title>Biorthogonal Wilson bases</title>

Cited by 10 publications

References 1 publication

Linear Approximation and Reproduction of Polynomials by Wilson Bases

Linear Approximation and Reproduction of Polynomials by Wilson Bases

Spline Modulation of Sinusoids for Signal Representation

Wilson basis expansions of electromagnetic wavefields: a suitable framework for fiber optics

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