Convolve a kernel {x,y} with a list of data:

Make a cyclic convolution the same length as the original data:

Align element 2 in the kernel with successive elements in the data:

Pad with zzz instead of using the data cyclically:

Two-dimensional convolution:

ListConvolve works with sparse arrays:

Use f in place of Times:

Use g in place of Plus:

Use f and g in place of Times and Plus, with empty data padding:

ListConvolve works with TimeSeries:

Find a moving average:

Or use the MovingAverage function:

Smooth noisy data:

Find the autocorrelation of a list:

Apply a simple image processing filter:

Multiply polynomials by convolving coefficient lists:

Multiply numbers by convolving digit lists:

Pascal's triangle:

Additive cellular automaton in base 5:

Fast multiplication of high‐degree polynomials:

Use ListConvolve with maximal overhangs and zero padding:

Generate two random vectors:

A function for constructing a circulant matrix from a vector:

Cyclic convolution is equivalent to multiplication with a circulant matrix:

Cyclic convolution is also equivalent to multiplication in the discrete Fourier transform domain:

Generate two random vectors:

A function for constructing a lower triangular Toeplitz matrix from a vector:

Cyclic convolution with zero-padding is equivalent to multiplication with a lower triangular Toeplitz matrix:

Convolve with a single element:

A kernel of the same length as the data, with no overhangs, is like a reversed dot product:

Successive differences:

Align with successive elements:

Varying alignments:

Varying alignments, with padding:

By default, the output is not the same length as the input:

Use overhangs to make the output the same length as the input:

Use overhangs to do the same in 2D: