EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/math/doc/interpolators/whittaker_shannon.qbk

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Copyright (c) 2019 Nick Thompson
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:whittaker_shannon Whittaker-Shannon interpolation]

[heading Synopsis]
``
  #include <boost/math/interpolators/whittaker_shannon.hpp>
``

  namespace boost { namespace math { namespace interpolators {

    template <class RandomAccessContainer>
    class whittaker_shannon
    {
    public:

        using Real = RandomAccessContainer::value_type;

        whittaker_shannon(RandomAccessContainer&& v, Real left_endpoint, Real step_size);

        Real operator()(Real x) const;

        Real prime(Real x) const;
    };

  }}} // namespaces


[heading Whittaker-Shannon Interpolation]

The Whittaker-Shannon interpolator takes equispaced data and interpolates between them via a sum of sinc functions.
This interpolation is stable and infinitely smooth, but has linear complexity in the data, 
making it slow relative to compactly-supported b-splines.
In addition, we cannot pass an infinite amount of data into the class, 
and must truncate the (perhaps) infinite sinc series to a finite number of terms.
Since the sinc function has slow /1/x/ decay, the truncation of the series can incur large error.
Hence this interpolator works best when operating on samples of compactly-supported functions.
Here is an example of interpolating a smooth "bump function":

    auto bump = [](double x) { if (std::abs(x) >= 1) { return 0.0; } return std::exp(-1.0/(1.0-x*x)); };

    double t0 = -1;
    size_t n = 2049;
    double h = 2.0/(n-1.0);

    std::vector<double> v(n);
    for(size_t i = 0; i < n; ++i) {
        double t = t0 + i*h;
        v[i] = bump(t);
    }

    auto ws = whittaker_shannon(std::move(v), t0, h);

    double y = ws(0.3);

The derivative of the interpolant can also be evaluated, but the accuracy is not as high:

    double yp = ws.prime(0.3);

[heading Complexity and Performance]

The call to the constructor requires [bigo](1) operations, simply moving data into the class.
Each call to the interpolant is [bigo](/n/), where /n/ is the number of points to interpolate.

[endsect] [/section:whittaker_shannon]