Spectral Functions for the Vector-Valued Fourier Transform
A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral...
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| Format: | Article |
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Wiley
2018-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2018/9584150 |
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| author | Vadim Mogilevskii |
| author_facet | Vadim Mogilevskii |
| author_sort | Vadim Mogilevskii |
| collection | DOAJ |
| description | A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}. |
| format | Article |
| id | doaj-art-84ada48e7aa5454499786d4f67ac2e72 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-84ada48e7aa5454499786d4f67ac2e722025-08-20T03:37:12ZengWileyJournal of Function Spaces2314-88962314-88882018-01-01201810.1155/2018/95841509584150Spectral Functions for the Vector-Valued Fourier TransformVadim Mogilevskii0Poltava V.G. Korolenko National Pedagogical University, Poltava, UkraineA scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.http://dx.doi.org/10.1155/2018/9584150 |
| spellingShingle | Vadim Mogilevskii Spectral Functions for the Vector-Valued Fourier Transform Journal of Function Spaces |
| title | Spectral Functions for the Vector-Valued Fourier Transform |
| title_full | Spectral Functions for the Vector-Valued Fourier Transform |
| title_fullStr | Spectral Functions for the Vector-Valued Fourier Transform |
| title_full_unstemmed | Spectral Functions for the Vector-Valued Fourier Transform |
| title_short | Spectral Functions for the Vector-Valued Fourier Transform |
| title_sort | spectral functions for the vector valued fourier transform |
| url | http://dx.doi.org/10.1155/2018/9584150 |
| work_keys_str_mv | AT vadimmogilevskii spectralfunctionsforthevectorvaluedfouriertransform |