The p-norm of circulant matrices via Fourier analysis
A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-01-01
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| Series: | Concrete Operators |
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| Online Access: | https://doi.org/10.1515/conop-2021-0123 |
| _version_ | 1797947736541102080 |
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| author | Sahasranand K. R. |
| author_facet | Sahasranand K. R. |
| author_sort | Sahasranand K. R. |
| collection | DOAJ |
| description | A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived. |
| first_indexed | 2024-04-10T21:31:29Z |
| format | Article |
| id | doaj.art-3698e63dde3441c1b66fa96f09e01092 |
| institution | Directory Open Access Journal |
| issn | 2299-3282 |
| language | English |
| last_indexed | 2024-04-10T21:31:29Z |
| publishDate | 2022-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Concrete Operators |
| spelling | doaj.art-3698e63dde3441c1b66fa96f09e010922023-01-19T13:20:29ZengDe GruyterConcrete Operators2299-32822022-01-01911510.1515/conop-2021-0123The p-norm of circulant matrices via Fourier analysisSahasranand K. R.0Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru560012, India.A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.https://doi.org/10.1515/conop-2021-0123self-adjointunitary invarianceinduced normriesz-thorin interpolation15b0547a30 |
| spellingShingle | Sahasranand K. R. The p-norm of circulant matrices via Fourier analysis Concrete Operators self-adjoint unitary invariance induced norm riesz-thorin interpolation 15b05 47a30 |
| title | The p-norm of circulant matrices via Fourier analysis |
| title_full | The p-norm of circulant matrices via Fourier analysis |
| title_fullStr | The p-norm of circulant matrices via Fourier analysis |
| title_full_unstemmed | The p-norm of circulant matrices via Fourier analysis |
| title_short | The p-norm of circulant matrices via Fourier analysis |
| title_sort | p norm of circulant matrices via fourier analysis |
| topic | self-adjoint unitary invariance induced norm riesz-thorin interpolation 15b05 47a30 |
| url | https://doi.org/10.1515/conop-2021-0123 |
| work_keys_str_mv | AT sahasranandkr thepnormofcirculantmatricesviafourieranalysis AT sahasranandkr pnormofcirculantmatricesviafourieranalysis |