Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes
In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function <i>f</i>, supposing to know only the samples of <i>f</i> at equidistant points. As reference interval we consider <inlin...
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MDPI AG
2020-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/4/542 |
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doaj-8f5b84032d0a486fa79cee542550b06f2020-11-25T02:28:54ZengMDPI AGMathematics2227-73902020-04-01854254210.3390/math8040542Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced NodesFrank Filbir0Donatella Occorsio1Woula Themistoclakis2Department of Scientific Computing, Helmholtz Zentrum München German Research Center for Environmental Health, Ingolstädter Landstrasse 1, 85764 Neuherberg, GermanyDepartment of Mathematics, Computer Science and Economics, University of Basilicata, viale dell’Ateneo Lucano 10, 85100 Potenza, ItalyC.N.R. National Research Council of Italy, IAC Institute for Applied Computing “Mauro Picone”, via P. Castellino, 111, 80131 Napoli, ItalyIn the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function <i>f</i>, supposing to know only the samples of <i>f</i> at equidistant points. As reference interval we consider <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results.https://www.mdpi.com/2227-7390/8/4/542Hilbert transformHadamard transformhypersingular integralBernstein polynomialsBoolean sumsimultaneous approximation |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Frank Filbir Donatella Occorsio Woula Themistoclakis |
| spellingShingle |
Frank Filbir Donatella Occorsio Woula Themistoclakis Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes Mathematics Hilbert transform Hadamard transform hypersingular integral Bernstein polynomials Boolean sum simultaneous approximation |
| author_facet |
Frank Filbir Donatella Occorsio Woula Themistoclakis |
| author_sort |
Frank Filbir |
| title |
Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes |
| title_short |
Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes |
| title_full |
Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes |
| title_fullStr |
Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes |
| title_full_unstemmed |
Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes |
| title_sort |
approximation of finite hilbert and hadamard transforms by using equally spaced nodes |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2020-04-01 |
| description |
In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function <i>f</i>, supposing to know only the samples of <i>f</i> at equidistant points. As reference interval we consider <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results. |
| topic |
Hilbert transform Hadamard transform hypersingular integral Bernstein polynomials Boolean sum simultaneous approximation |
| url |
https://www.mdpi.com/2227-7390/8/4/542 |
| work_keys_str_mv |
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