Wavelet Transform and Filter Banks
Charles K. Chui, Qingtang Jiang · Applied Mathematics · 2013
To adopt the Fourier basis functions $$e^{i \omega x}$$ or $$e^{i n x}, \cos n x$$ , and $$ \sin n x$$ , where $$\omega \in \mathbb R $$ and $$n \in \mathbb Z $$ , for investigating the frequency contents of functions $$f (x)$$ , for $$ f \in L_1 (\mathbb R )$$ or $$f \in PC^* [a,b]$$ , in some desirable neighborhood of any $$x = x_0$$ , called “region of interest (ROI)” in engineering applications, a suitable window function is used to introduce the localized Fourier transform (LFT) and to construct time-frequency basis functions in Sects. 7.4 and 7.5 , respectively, of the previous chapter.