This Blog is Mainly for those who are doing Full Fledged Research in Wireless Communications and Signal Processing,Control Systems, Power Systems, Chemical Engg,Computer Science,etc .Math Prequisites for this include Calculus, Differential Equations,Probability and Stochastic Process,Signal Analysis,Linear Algebra,Optimization,Real and Complex Analysis, Information Theory and List Goes on...I will be Posting some Interesting Problems in these Math Fields and Discussions are Most Welcomed :)

HTML Code

Thursday, 18 April 2013

Functions

Can we Find any two Distinct Functions
$$ f:\Re \rightarrow \Re $$ such that

$$ f\left ( x \right )=f\left (\frac{x}{2}\right ) $$

It's easy to see that formally speaking $$L(x/2) = L(x)$$. Now that series I gave seems hopeless at first since it consider values $m$ at 0 and infinity. That is fine, let us consider $m(x) = x^3e^{-x^2}$. This rapidly goes to 0 as the $|x|$ gets very large, as well as very small. Then substituting that back into the definition of $L$ gives you a function that is periodic with respect to doubling the period (at least over the reals).

If it is true for all x, then I think it would be f(x)=0.

ReplyDeleteIn general $$f(x)=Const$$ is one such function.

ReplyDeleteWe can engineer a solution pretty easily here. Consider some $m(x)$ (we'll talk about what this is later).

ReplyDeleteConsider

$$L(x)= m(4x) + m(2x) + m(x) + m(\frac{x}{2}) + m(\frac{x}{8}) + ... $$

It's easy to see that formally speaking $$L(x/2) = L(x)$$. Now that series I gave seems hopeless at first since it consider values $m$ at 0 and infinity. That is fine, let us consider $m(x) = x^3e^{-x^2}$. This rapidly goes to 0 as the $|x|$ gets very large, as well as very small. Then substituting that back into the definition of $L$ gives you a function that is periodic with respect to doubling the period (at least over the reals).

The $L(x)$ in my earlier comment should be $L(x) = ... m(4x) ... $ that is, it is infinite in both directions

ReplyDelete