Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals
The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited s...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2012-11-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | http://www.mdpi.com/1099-4300/14/11/2192 |
| _version_ | 1818035756377047040 |
|---|---|
| author | Rudolf L. Stens Gerhard Schmeisser Paul L. Butzer |
| author_facet | Rudolf L. Stens Gerhard Schmeisser Paul L. Butzer |
| author_sort | Rudolf L. Stens |
| collection | DOAJ |
| description | The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications. |
| first_indexed | 2024-12-10T07:00:07Z |
| format | Article |
| id | doaj.art-6a082e61a8ed47ffbc797cd99642f963 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-12-10T07:00:07Z |
| publishDate | 2012-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-6a082e61a8ed47ffbc797cd99642f9632022-12-22T01:58:21ZengMDPI AGEntropy1099-43002012-11-0114112192222610.3390/e14112192Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited SignalsRudolf L. StensGerhard SchmeisserPaul L. ButzerThe paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.http://www.mdpi.com/1099-4300/14/11/2192sampling formulaedifferentiation formulaenon-bandlimited functionsaliasing errorHilbert transformsformulae with remaindersderivative-free error estimatesBernstein’s inequality |
| spellingShingle | Rudolf L. Stens Gerhard Schmeisser Paul L. Butzer Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals Entropy sampling formulae differentiation formulae non-bandlimited functions aliasing error Hilbert transforms formulae with remainders derivative-free error estimates Bernstein’s inequality |
| title | Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals |
| title_full | Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals |
| title_fullStr | Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals |
| title_full_unstemmed | Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals |
| title_short | Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals |
| title_sort | shannon s sampling theorem for bandlimited signals and their hilbert transform boas type formulae for higher order derivatives the aliasing error involved by their extensions from bandlimited to non bandlimited signals |
| topic | sampling formulae differentiation formulae non-bandlimited functions aliasing error Hilbert transforms formulae with remainders derivative-free error estimates Bernstein’s inequality |
| url | http://www.mdpi.com/1099-4300/14/11/2192 |
| work_keys_str_mv | AT rudolflstens shannonssamplingtheoremforbandlimitedsignalsandtheirhilberttransformboastypeformulaeforhigherorderderivativesthealiasingerrorinvolvedbytheirextensionsfrombandlimitedtononbandlimitedsignals AT gerhardschmeisser shannonssamplingtheoremforbandlimitedsignalsandtheirhilberttransformboastypeformulaeforhigherorderderivativesthealiasingerrorinvolvedbytheirextensionsfrombandlimitedtononbandlimitedsignals AT paullbutzer shannonssamplingtheoremforbandlimitedsignalsandtheirhilberttransformboastypeformulaeforhigherorderderivativesthealiasingerrorinvolvedbytheirextensionsfrombandlimitedtononbandlimitedsignals |