Mixed thoughts on Wavelets

Daubechies wavelets: Family of orthogonal wavelets with smallest support for a given “number of vanishing moments” –> good representation properties for piecewise polynomials
Symlets are the “least asymmetric” variant: Slightly less regular but the filter with a near-linear phase results in more symmetric wavelets

Riesz basis: generalization of orthonormal bases (all orthonormal bases are Riesz bases)
$l^p$ space: space of all sequences satisfying $\sum_n \vert x_n \vert_p < \infty$
-> Sum of norms is finite.
example: $l^2$ is all square summable sequences
$L^p$ space: same with functions, e.g. $L^2$ is space of all square integrateable functions (= finite energy), called “Lebesgue space”
Sobolev space $W^k_p$: Subset of functions $f \in L^p$ whose weak derivatives up to order $k$ have finite $L^p$ norm
Important special case: $p=2$: Forms Hilbert space, short notation often $W^k_2 = H^k$
Schauder basis: countable basis (can have infinite sum to represent element instead of finite in ordinary Hamel basis)

Monoic polynomial: polynomial with leading coefficient (highest order) equal to 1
Hyperbolic wavelet basis: linear approximation via “sparse grid-type bases”
Smoothness: $C^n$ means function has $n$ continous derivatives
Parseval’s theorem: analogue of Pythagoras’ theorem for functions and proofs completeness (see notebook)

$C^n$ is continuity:
- $C^1$: continuity of the tangent vector: “continuous derivative”
- $C^2$: continuity of the slope
Knots: how many points “coalesce”
- simple knot: knot with $C^n-1$ continuity
- double knot: knot with $C^n-2$ continuity