= Solution
With $N$ <vanishing moments>, a <wavelet> annihilates every <polynomial> of degree below $N$. A <Taylor polynomial> then explains small <wavelet> coefficients on smooth parts of a signal, and efficient <best N-term approximation>. <Compact support> localizes coefficients near a feature, limits the number of boundary interactions, and permits a finite filter implementation. Increasing the number of <vanishing moments> while keeping an <orthonormal basis> generally requires a larger <support of a function>: a finite orthonormal filter with $N$ <vanishing moments> needs at least $2N$ taps, and the minimal-support <Daubechies wavelet> has support length $2N-1$ in the standard normalization.
The <Haar wavelet>, $\psi=\chi_{[0,1/2)}-\chi_{[1/2,1)}$, has one <vanishing moment>, unit support length and discontinuities. It is inexpensive and particularly suitable for piecewise constant data with sharp jumps. A <Daubechies wavelet> of a larger order has more <vanishing moments> and a longer finite filter; sufficiently large orders also provide greater regularity. It is useful when smooth trends should yield small coefficients, although the wider <support of a function> can spread a jump across more coefficients. \b[Choose Haar for compact jump localization; choose a higher-order Daubechies wavelet for smooth polynomial structure.] More <vanishing moments> alone does not make every low-order wavelet highly differentiable.
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