Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. Also let $\mathbf{x}=(x_1,\ldots,x_d)\in\mathbb{R}^d$.

I'd like to prove the following:

If

$$\int_{\mathbb{R}^d}x_k\exp(-\mathbf{x}^T\mathbf{x})f(g(x))\,d\mathbf{x}=0$$

for all bounded continuous functions $f$, then $g(\mathbf{x})=g(-\mathbf{x})$ for all $\mathbf{x}\in\mathbb{R}^d$, that is $g$ is an even function.

share|improve this question
1  
Something is missing here, I think. Right now you are integrating a vector-valued function - perhaps your integrand is supposed to be $\vert\textbf{x}\vert\exp(-\textbf{x}^\intercal\textbf{x})f(g(\textbf{x}))$? –  icurays1 Jan 24 '13 at 19:06
2  
Presumably the integral is the vector valued integral, ie, $[\int_{\mathbb{R}^d}\mathbf{x}\exp(-\mathbf{x}^T\mathbf{x})f(g(x))\,d\mathbf{x}‌​]_k = \int_{\mathbb{R}^d}\mathbf{x_k}\exp(-\mathbf{x}^T\mathbf{x})f(g(x))\,d\mathbf{x}‌​$. –  copper.hat Jan 24 '13 at 19:31
    
Sorry I meant what copper.hat has written. Thanks for spotting the mistake! –  red271 Jan 25 '13 at 8:24

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.