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[/
Copyright (c) 2019 Nick Thompson
Copyright (c) 2019 Paul A. Bristow
Use, modification and distribution are subject to the
Boost Software License, Version 1.0. (See accompanying file
LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
]
[section:wavelet_transforms Wavelet Transforms]
[heading Synopsis]
```
#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 \u03C8 is
[$../graphs/wavelet_transform_definition.svg]
For compactly supported Daubechies wavelets, the bounds can always be taken as finite, and we have
[$../graphs/daubechies_wavelet_transform_definition.svg]
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.
[$../graphs/scalogram_sin1t_light.png]
[endsect] [/section:wavelet_transforms]
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