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Fourier series is a useful orthonormal representation on ${L}^{2}(\left[0 , T\right])$ especiallly for inputs into LTI systems. However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena ).
Wavelets , discovered in the last 15 years, are another kind of basis for ${L}^{2}(\left[0 , T\right])$ and have many nice properties.
Fourier series - ${c}_{n}$ give frequency information. Basis functions last the entire interval.
Wavelets - basis functions give frequency info but are local in time.
In Fourier basis, the basis functions are harmonic multiples of $e^{i{\omega}_{0}t}$
In Haar wavelet basis , the basis functions are scaled and translated versions of a "mother wavelet" $\psi (t)$ .
Basis functions $\{{\psi}_{j,k}(t)\}$ are indexed by a scale j and a shift k.
Let $\forall , 0\le t< T\colon \phi (t)=1$ Then $\{\phi (t)2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)\colon j\in \mathbb{Z}\land (k=0,1,2,\dots ,{2}^{j}-1)\}$
Let ${\psi}_{j,k}(t)=2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)$
Larger $j$ → "skinnier" basis function, $j=\{0, 1, 2, \dots \}$ , $2^{j}$ shifts at each scale: $k=0,1,\dots ,{2}^{j}-1$
Check: each ${\psi}_{j,k}(t)$ has unit energy
Any two basis functions are orthogonal.
Also, $\{{\psi}_{j,k}, \phi \}$ span ${L}^{2}(\left[0 , T\right])$
Using what we know about Hilbert spaces : For any $f(t)\in {L}^{2}(\left[0 , T\right])$ , we can write
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