Eigenvalues convolution-type operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:40:32Z http://mathoverflow.net/feeds/question/39011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39011/eigenvalues-convolution-type-operator Eigenvalues convolution-type operator Jonas Teuwen 2010-09-16T19:28:01Z 2010-09-16T21:14:11Z <p>Let $J_1$ be the Bessel function of the first kind and let $H_1(x) = \frac{J_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f * H_1)(x)$ from $L^2$ to $L^2$.</p> <p>Since the $H_1$-function is the Fourier transform of something it must be in $L^2$, so we have a Hilbert-Schmidt operator which is in this case self-adjoint and compact, so the spectral theorem applies which for example says all the eigenvalues form a countable set, are real and go to zero.</p> <p>What I am now interested in is the largest eigenvalue of $T$. What theorem or method could I try to obtain this? (I don't need a full solution, just a hint would suffice).</p> http://mathoverflow.net/questions/39011/eigenvalues-convolution-type-operator/39021#39021 Answer by Helge for Eigenvalues convolution-type operator Helge 2010-09-16T20:53:00Z 2010-09-16T20:53:00Z <p>The easy way to deal with convolution operators $f \mapsto g\ast f$ is to Fourier-transform. Then the operator is just the multiplication by $\hat{g}$ and the spectrum is the essential range of $\hat{g}$. Of course this is not an eigenvalue, these all live on the constancy intervals of $\hat{g}$, so just determine these. In your edit these are $1$ and $0$ for the circ function.</p> http://mathoverflow.net/questions/39011/eigenvalues-convolution-type-operator/39022#39022 Answer by Julián Aguirre for Eigenvalues convolution-type operator Julián Aguirre 2010-09-16T21:14:11Z 2010-09-16T21:14:11Z <p>In Fourier space, the operator $T$ is given by $$\hat{(T f)}(\xi)=c\sqrt{1-\xi^2}\hat f(\xi)$$ for some constant $c\ne0$ ($c$ dependson the normalization chosen for thr Fourier transform.) If $\lambda$ is an eigenvalue and $f$ is a corresponding eigenfunction, then for almost all $\xi$ $$\lambda \hat f(\xi)=c\sqrt{1-\xi^2}\hat f(\xi).$$ It follows that the only eigenvalue is $\lambda=0$, and the coresponding eigenfunctions are all $L^2$ functions whose Fourier transform vanish almost everywhere on $[-1,1]$.</p>