Chirps Construct on Sobolev Spaces and the Behavior of their ${\left\|\hspace{.5cm}\right\|}_{L^2[-\varepsilon,\varepsilon]}$ and ${\left\|\hspace{.5cm}\right\|}_{H^s[-\varepsilon ,\varepsilon ]}$ Norm

Abstract

The main idea of this work is to characterize chirps construct on Sobolev spaces by studying the behavior of ${\left\|f\right\|}_{L^2[-\varepsilon ,\varepsilon ]}$ or ${\left\|f\right\|}_{H^s[-\varepsilon ,\varepsilon ]}$. We expect that the behavior is in order of ${\varepsilon }^{\alpha +\left(\beta +1\right)(\left|s\right|+\frac{1}{2})}$ when $s$ tend to $-\infty $, and we believe that we have equivalence between this and the definition of chirps construct on Sobolev spaces. The formula of a chirp is given in the form ${f\left(x\right)=\left|x\right|}^{\alpha }g({\left|x\right|}^{-\beta })$, where $\beta >0$ and $g$ is an indefinitely oscillating function on $L^2$ or $H^s$. Which means that $g$ has for all $m$ integer one primitive of the order $m$ in the same space.