For 10 points, name this binary operation that maps functions f and g to "integral f of x times g of quantity u minus x dx" and is denoted "f star g".
Gelfand's problem of describing all closed invariant subspaces of the integration operator may be solved using the Titchmarsh theorem of this type. This operation is often represented as a Toeplitz matrix, and the Dirichlet variety of this operation for a given integer n is defined using a sum over all divisors of n. Its discrete analog is naturally computed in "big O of N squared" with respect to resolution, but can be sped by switching to a circular domain of summation and using a namesake theorem to transform the problem to a product of fast Fourier transforms; this enables image processing applications like edge detection and modeling distortions such as blurring. For 10 points, name this binary operation that maps functions f and g to "integral f of x times g of quantity u minus x dx" and is denoted "f star g".
Gelfand's problem of describing all closed invariant subspaces of the integration operator may be solved using the Titchmarsh theorem of this type.
convolution