
Math General Math Forum  For general math related discussion and news 
 LinkBack  Thread Tools  Display Modes 
July 15th, 2018, 07:40 PM  #1 
Newbie Joined: Jun 2018 From: Viet Nam Posts: 3 Thanks: 0  Prove the eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(xy)\phi_j(y)dy}$ is
Prove the eigenvalues $\lambda$ of $\lambda \phi_j(x)= \int_G{ K(xy)\phi_j(y)dy}$ 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 2Rperodic. When I try to devide the convolution by $\phi_j(x)$, I have $\lambda=\int_G{K(xy)\phi_j(y) / \phi_j(x)dy}=(2R)^{n/2}\int_G{K(xy)\phi_j(yx)dy}$. Let $t=xy$ 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, 01:52 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,642 Thanks: 627  

Tags 
$lambda, $lambda$, eigenvalues, intg, kxyphijydy$, phijx, prove 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Lambda Calc  Kujanator  Computer Science  12  April 20th, 2015 05:33 AM 
Defining a new symmetry group  n dimensional permutation invariance of lambda  BenFRayfield  Computer Science  0  February 12th, 2015 02:01 AM 
Lambda Tensor  a minimalist general computing math operator  BenFRayfield  Linear Algebra  2  March 8th, 2013 07:57 PM 
Properties of Lambda Operator in Schwarzschild equation  123Peter  Calculus  0  February 5th, 2011 05:09 AM 
limit of this integral as lambda goes to infinity  APK  Real Analysis  2  October 15th, 2009 04:04 AM 