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 July 15th, 2018, 06:40 PM #1 Newbie   Joined: Jun 2018 From: Viet Nam Posts: 3 Thanks: 0 Prove the eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(x-y)\phi_j(y)dy}$ is Prove the eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(x-y)\phi_j(y)dy}$ is $\int_G{K(x)\phi_{-j}(x)dx}$, with $\phi_j(x)=(2R)^{-n/2}exp(i\pi j. \frac{x}{R}), j \in \mathbb{Z}^n, x, y \in \mathbb{R}^n, G=\{x \in \mathbb{R}^n: |x_i|\leq R,i=1,...,n\}$ and $K(x)$ is 2R-perodic. When I try to devide the convolution by $\phi_j(x)$, I have $\lambda=\int_G{K(x-y)\phi_j(y) / \phi_j(x)dy}=(2R)^{n/2}\int_G{K(x-y)\phi_j(y-x)dy}$. Let $t=x-y$ assume that $t\in G$, so $dt=-dy$ and $\lambda= - (2R)^{n/2}\int_{G}{K(t)\phi_{-j}(t)dt}$. What's wrong with me? July 16th, 2018, 12:52 PM #2 Global Moderator   Joined: May 2007 Posts: 6,707 Thanks: 674 You'll have better luck with mathematics stack exchange https://math.stackexchange.com/ Tags $lambda,$lambda$, eigenvalues, intg, kxyphijydy$, phijx, prove Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Kujanator Computer Science 12 April 20th, 2015 04:33 AM BenFRayfield Computer Science 0 February 12th, 2015 01:01 AM BenFRayfield Linear Algebra 2 March 8th, 2013 06:57 PM 123Peter Calculus 0 February 5th, 2011 04:09 AM APK Real Analysis 2 October 15th, 2009 03:04 AM

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