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 
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