...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

#include <boost/math/quadrature/wavelet_transforms.hpp> namespace boost::math::quadrature { template<class F, typename Real, int p> class daubechies_wavelet_transform { public: daubechies_wavelet_transform(F f, int grid_refinements = -1, Real tol = 100*std::numeric_limits<Real>::epsilon(), int max_refinements = 12) {} daubechies_wavelet_transform(F f, boost::math::daubechies_wavelet<Real, p> wavelet, Real tol = 100*std::numeric_limits<Real>::epsilon(), int max_refinements = 12); auto operator()(Real s, Real t)->decltype(std::declval<F>()(std::declval<Real>())) const; }; }

The wavelet transform of a function *f* with respect to
a wavelet ψ is

For compactly supported Daubechies wavelets, the bounds can always be taken as finite, and we have

which also defines the *s*=0 case.

The code provided by Boost merely forwards a lambda to the trapezoidal quadrature routine, which converges quickly due to the Euler-Maclaurin summation formula. However, the convergence is not as rapid as for infinitely differentiable functions, so the default tolerances are modified.

A basic usage is

auto psi = daubechies_wavelet<double, 8>(); auto f = [](double x) { return sin(1/x); }; auto Wf = daubechies_wavelet_transform(f, psi); double w = Wf(0.8, 7.2);

An image from this function is shown below.