# 7.3 Haar approximation at the kth coarseness level

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It is instructive to consider the approximation of signal $x(t)\in {ℒ}_{2}$ at coarseness-level- $k$ of the Haar system. For the Haar case, projection of $x(t)\in {ℒ}_{2}$ onto ${V}_{k}$ is accomplished using the basis coefficients

${c}_{k,n}=\int_{()} \,d t$ φ k , n t x t t n 2 k n 1 2 k 2 k 2 x t
giving the approximation
${x}_{k}(t)=\sum_{n=()}$ c k , n φ k , n t n t n 2 k n 1 2 k 2 k 2 x t φ k , n t n 1 2 k t n 2 k n 1 2 k x t 2 k 2 φ k , n t
where $\frac{1}{2^{k}}\int_{n2^{k}}^{(n+1)2^{k}} x(t)\,d t=\text{average value of x(t) in interval}$ $\forall k\colon 2^{\left(\frac{k}{2}\right)}{\phi }_{k,n}(t)=\text{height}=1$ This corresponds to taking the average value of the signal in each interval of width $2^{k}$ and approximating the function by a constant over that interval (see ).

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I think
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