Let nonequispaced nodes
and frequencies
in the index sets
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The nonequispaced discrete cosine and sine transforms are given by
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The straight forward algorithm of this matrix vector product, which is called
ndct and ndst, takes
and
arithmetical operations.
For these real transforms the adjoint transforms coincide with the ordinary
transposed matrix vector products.
Our fast approach is based on the NFFT and seems to be easier than the
Chebyshev transform based derivation in [44] and faster than the
algorithms in [56] which still use FFTs.
Instead of FFTs we use fast algorithms for the discrete cosine transform
(DCT-I) and for the discrete sine transform (DST-I).
For details we refer to [22].