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


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Authors

  • T. Elbouayachi D\`{e}partement de Math\'{e}matiques, Universit\'{e} Abdelmalek Essaadi, FST Tanger, Tanger, Morocco

Keywords:

Sobolev Spaces, Norm, Behavior of ${\left\|f\right\|}_{L^2[-\varepsilon ,\varepsilon ]}$

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.

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Published

15-03-2019

How to Cite

T. Elbouayachi. (2019). 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. International Journal of Mathematics And Its Applications, 7(1), 31–39. Retrieved from https://ijmaa.in/index.php/ijmaa/article/view/266

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Section

Research Article