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I want to perform convolution for a sequence but instead of having the indices cycle from the ends periodically, I want to apply a function, specifically Abs, to remap the indices.

Using ListConvolve, what parameters and/or modifications to the inputs could allow this output?

For example, here is my specific case. My input sequences will have equal length. Using a summation, I can obtain the results I desire, but this runs in $O(n^2)$.

For an arbitrary sequence of length 5, this would be the convolution using summation.

Table[Sum[Subscript[a, j] Subscript[b, Abs[k - j] + 1], {j, 5}], {k, 5}]
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  • $\begingroup$ Can't this be expressed as a conventional convolution with a symmetric sequence b? $\endgroup$ – mikado Jun 19 '16 at 10:19
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How about

k = Table[Subscript[a, j] , {j, 5}];    
l = Table[Subscript[b, 1 + Abs[j]] , {j, -4, 4}];

ListCorrelate[k, l] // Reverse // Column

enter image description here

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