A proof of the definition of Haar functions.

by weakmathematician   Last Updated May 27, 2017 10:20 AM

Let $I = I_{j,k}$ be a dyadic interval. The Haar function associated with I is defined by$$h_{j,k}(x) =\frac{1}{|I|} (\chi_{I_{R}(x)} - \chi_{I_{L}(x)} ),$$ where, $I_{R} = [(2k + 1)2^{-j-1}, (k+1)2^{-j}[$, $I_{L} = [k2^{-j}, (2k+1)2^{-j-1}[.$

Show that $$h_{j,k}(x) = 2^{j/2} h_{0,0}(2^{j}x - k).$$

Could anyone give me a hint?

Answers 1

Note that $$h_{0,0}(x)=\begin{cases}-1 & 0≤x<\frac12\\ 1& \frac12≤x<1\end{cases}\qquad |I|\cdot h_{i,j}(x)=\begin{cases}-1 & k≤2^jx<k+\frac12\\ 1& k+\frac12≤2^jx<k+1\end{cases}$$ where I would usually say $|I|=2^{-j}$, but your result needs $|I|=2^{-j/2}$. At any rate compare the two expressions by using the explicit form of $h$ that I wrote out before.

May 27, 2017 12:38 PM

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