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A generalized Walsh system for arbitrary matrices
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| Zeitschriftentitel: | Demonstratio Mathematica |
|---|---|
| Personen und Körperschaften: | , |
| In: | Demonstratio Mathematica, 52, 2019, 1, S. 40-55 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Walter de Gruyter GmbH
|
| Schlagwörter: |
| author_facet |
Harding, Steven N. Picioroaga, Gabriel Harding, Steven N. Picioroaga, Gabriel |
|---|---|
| author |
Harding, Steven N. Picioroaga, Gabriel |
| spellingShingle |
Harding, Steven N. Picioroaga, Gabriel Demonstratio Mathematica A generalized Walsh system for arbitrary matrices General Mathematics |
| author_sort |
harding, steven n. |
| spelling |
Harding, Steven N. Picioroaga, Gabriel 2391-4661 Walter de Gruyter GmbH General Mathematics http://dx.doi.org/10.1515/dema-2019-0006 <jats:title>Abstract</jats:title> <jats:p> In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.</jats:p> A generalized Walsh system for arbitrary matrices Demonstratio Mathematica |
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Walter de Gruyter GmbH, 2019 |
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A generalized Walsh system for arbitrary matrices |
| title_unstemmed |
A generalized Walsh system for arbitrary matrices |
| title_full |
A generalized Walsh system for arbitrary matrices |
| title_fullStr |
A generalized Walsh system for arbitrary matrices |
| title_full_unstemmed |
A generalized Walsh system for arbitrary matrices |
| title_short |
A generalized Walsh system for arbitrary matrices |
| title_sort |
a generalized walsh system for arbitrary matrices |
| topic |
General Mathematics |
| url |
http://dx.doi.org/10.1515/dema-2019-0006 |
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2019 |
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40-55 |
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<jats:title>Abstract</jats:title>
<jats:p> In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.</jats:p> |
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| author | Harding, Steven N., Picioroaga, Gabriel |
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| description | <jats:title>Abstract</jats:title> <jats:p> In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.</jats:p> |
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| imprint_str_mv | Walter de Gruyter GmbH, 2019 |
| institution | DE-Ch1, DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161, DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14 |
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| series | Demonstratio Mathematica |
| source_id | 49 |
| spelling | Harding, Steven N. Picioroaga, Gabriel 2391-4661 Walter de Gruyter GmbH General Mathematics http://dx.doi.org/10.1515/dema-2019-0006 <jats:title>Abstract</jats:title> <jats:p> In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.</jats:p> A generalized Walsh system for arbitrary matrices Demonstratio Mathematica |
| spellingShingle | Harding, Steven N., Picioroaga, Gabriel, Demonstratio Mathematica, A generalized Walsh system for arbitrary matrices, General Mathematics |
| title | A generalized Walsh system for arbitrary matrices |
| title_full | A generalized Walsh system for arbitrary matrices |
| title_fullStr | A generalized Walsh system for arbitrary matrices |
| title_full_unstemmed | A generalized Walsh system for arbitrary matrices |
| title_short | A generalized Walsh system for arbitrary matrices |
| title_sort | a generalized walsh system for arbitrary matrices |
| title_unstemmed | A generalized Walsh system for arbitrary matrices |
| topic | General Mathematics |
| url | http://dx.doi.org/10.1515/dema-2019-0006 |