Nonnegative Fourier series coefficients for periodic nonnegative-definite function
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Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $Sigma_{ij}=kappa(x_i-x_j)$ is nonnegative-definite for any sequence of real numbers $x_1,ldots,x_n$. AKA positive semi-definite.)
It seems that some version of Bochner's Theorem ought to do it, but I'm having trouble finding the version appropriate to periodic functions.
fourier-series positive-semidefinite
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
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add a comment |
$begingroup$
Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $Sigma_{ij}=kappa(x_i-x_j)$ is nonnegative-definite for any sequence of real numbers $x_1,ldots,x_n$. AKA positive semi-definite.)
It seems that some version of Bochner's Theorem ought to do it, but I'm having trouble finding the version appropriate to periodic functions.
fourier-series positive-semidefinite
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
$endgroup$
add a comment |
$begingroup$
Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $Sigma_{ij}=kappa(x_i-x_j)$ is nonnegative-definite for any sequence of real numbers $x_1,ldots,x_n$. AKA positive semi-definite.)
It seems that some version of Bochner's Theorem ought to do it, but I'm having trouble finding the version appropriate to periodic functions.
fourier-series positive-semidefinite
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
$endgroup$
Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $Sigma_{ij}=kappa(x_i-x_j)$ is nonnegative-definite for any sequence of real numbers $x_1,ldots,x_n$. AKA positive semi-definite.)
It seems that some version of Bochner's Theorem ought to do it, but I'm having trouble finding the version appropriate to periodic functions.
fourier-series positive-semidefinite
fourier-series positive-semidefinite
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
asked Dec 10 '18 at 22:00
Kevin S. Van HornKevin S. Van Horn
11
11
add a comment |
add a comment |
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