g matrix
of coefficients orthonormal (``O @ O.T = np.eye(N)``).

The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up
to a factor :math:`2N`. The orthonormalized DST-III is exactly the inverse of the
orthonormalized DST-II.

**Type IV**

There are several definitions of the DST-IV, we use the following (for
``norm="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and
even around :math:`n=N-0.5`

.. math::

    y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)

``orthogonalize`` has no effect here, as the DST-IV matrix is already
orthogonal up to a scale factor of ``2N``.

The (unnormalized) DST-IV is its own inverse, up to a factor :math:`2N`. The
orthonormalized DST-IV is exactly its own inverse.

References
----------
.. [1] Wikipedia, "Discrete sine transform",
       https://en.wikipedia.org/wiki/Discrete_sine_transform

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