Trigonometric interpolation
The approximate representation of a function in the form of a trigonometric polynomial
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whose values coincide at prescribed points with the corresponding values of the function. Thus, it is always possible to choose the coefficients
,
,
,
, of the
-th order polynomial
so that its values are equal to the values
of the function at
preassigned points
in the interval
. The polynomial has the form
![]() | (*) |
where
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The polynomial assumes an especially simple form in case the nodes
are equi-distant; the coefficients are given by the formulas
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Comments
The formula (*) above for the trigonometric polynomial taking the prescribed values at the nodes
is known as the Gauss formula of trigonometric interpolation, [a2].
References
[a1] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) |
[a2] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 29, 38 |
Trigonometric interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_interpolation&oldid=38916