We may consider orthogonal functions, and posit a problem of finding a basis in such a space. The quest for bases in function spaces with scalar product led to the theory of Fourier series and Fourier transform, to the theory of generalized functions and most recently to the discovery of wavelets . As in the finite-dimensional case, scalar product is not a group operation because its result does not belong to the space itself. These are reflected shifts of the function f . This function is known as the convolution of f and g .
Convolutions arise naturally when differential equations are being replaced with integral ones.
Thus integral equations with convolutions may be further reduced to algebraic equations. The unit element for the convolution is the well-known Dirac delta function. Later S.L.Sobolyev and L.Schwartz developed a framework of generalized functions in which such a definition indeed made sense. Alexander Bogomolny 2497430 .