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The first problem to be addressed can be regarded as a matrix vector
multiplication. For a finite number of given Fourier coefficients
,
, we consider the evaluation of the
trigonometric polynomial
![$\displaystyle f\left(\ensuremath{\boldsymbol{x}}\right) := \sum_{\ensuremath{\b...
...\mbox{\scriptsize {i}}} \ensuremath{\boldsymbol{k}}\ensuremath{\boldsymbol{x}}}$](img54.png) |
(2.1) |
at given nonequispaced nodes
.
Thus, our concern is the evaluation of
![$\displaystyle f_j = f\left(\ensuremath{\boldsymbol{x}}_j\right) := \sum_{\ensur...
...ox{\scriptsize {i}}} \ensuremath{\boldsymbol{k}}\ensuremath{\boldsymbol{x}}_j},$](img56.png) |
(2.2) |
.
In matrix vector notation this reads
![$\displaystyle \ensuremath{\boldsymbol{f}}=\ensuremath{\boldsymbol{A}} \ensuremath{\boldsymbol{\hat f}}$](img58.png) |
(2.3) |
where
The straightforward algorithm for computing this matrix vector product,
which is called NDFT, takes
arithmetical operations.
A closely related matrix vector product is the adjoint NDFT
where
denotes the conjugate transpose
of the nonequispaced Fourier matrix
.
Subsections
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Up: Notation, the NDFT, and
Previous: Notation, the NDFT, and
Contents
Jens Keiner
2006-11-20