Built-in Mathematica Symbol

ListConvolve

ListConvolve[ker, list]
forms the convolution of the kernel ker with list.

ListConvolve[ker, list, k]
forms the cyclic convolution in which the k

element of ker is aligned with each element in list.

ListConvolve[ker, list, {k_L, k_R}]
forms the cyclic convolution whose first element contains list[[1]]ker[[k_L]] and whose last element contains list[[-1]]ker[[k_R]].

ListConvolve[ker, list, klist, p]
forms the convolution in which list is padded at each end with repetitions of the element p.

ListConvolve[ker, list, klist, {p₁, p₂, ...}]
forms the convolution in which list is padded at each end with cyclic repetitions of the p_i.

ListConvolve[ker, list, klist, padding, g, h]
forms a generalized convolution in which g is used in place of Times and h in place of Plus.

ListConvolve[ker, list, klist, padding, g, h, lev]
forms a convolution using elements at level lev in ker and list.

MORE INFORMATION

With kernel K_r and list a_s, ListConvolve[ker, list] computes , where the limits of the sum are such that the kernel never overhangs either end of the list.

ListConvolve[ker, list] gives a result of length Length[list]-Length[ker]+1.

ListConvolve[ker, list] allows no overhangs and is equivalent to ListConvolve[ker, list, {-1, 1}].

ListConvolve[ker, list, k] is equivalent to ListConvolve[ker, list, {k, k}].

The values of k_L and k_R in ListConvolve[ker, list, {k_L, k_R}] determine the amount of overhang to allow at each end of list.

Common settings for {k_L, k_R} are:

	{-1,1}	no overhangs (default)
	{-1,-1}	maximal overhang at the right-hand end
	{1,1}	maximal overhang at the left-hand end
	{1,-1}	maximal overhangs at both beginning and end

With maximal overhang at one end only, the result from ListConvolve is the same length as list.

ListConvolve[ker, list, {k_L, k_R}, padlist] effectively lays down repeated copies of padlist, then superimposes one copy of list on them and forms a convolution of the result.

Common settings for padlist are:

	p	pad with repetitions of a single element
	{p₁,p₂,...}	pad with cyclic repetitions of a sequence of elements
	list	pad by treating list as cyclic (default)
	{}	do no padding

ListConvolve works with multidimensional kernels and lists of data.

ListConvolve[ker, list, {{k_L1, k_L2, ...}, {k_R1, k_R2, ...}}] forms the cyclic convolution whose {1, 1, ...} element contains ker[[k_L1, k_L2, ...]]list[[1, 1, ...]] and whose {-1, -1, ...} element contains ker[[k_R1, k_R2, ...]]list[[-1, -1, ...]].