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First off, apologies if this is in the wrong place - any directions on where this is more appropriate are appreciated.

I am trying to figure out how to convert a smoothing operation in physical space into one in Fourier space.

In physical space, with equally spaced points [itex]x_j[/itex], (j =1,2,3...), we

**define**an averaging of a function [itex] f(x) [/itex] as

[tex] \bar{f}_j =\frac{1}{16}(-f_{j-2} +4 f_{j-1} + 10 f_j + 4f_{j+1} -f_{j+2})[/tex],

where [itex] f_j=f(x_j) [/itex]. The operation here is a type of moving average that smooths a function that has an oscillation, of period 2, about a smooth function.

Now, I do not have information in physical space, rather I have the fourier coefficients of f, i.e. I know the set [itex] \hat{f}_k, (k=1,2,...) [/itex], and I would like to apply the averaging to these coefficients.

I'm wondering if there's an analog for this spatial averaging in fourier space, but am very much out of my area of expertise.

Any tips, or even relevant references, would be much appreciated.

Nick