# Convolutional operators

Even if we work only with finite and discrete convolutions, it's useful to start providing the standard definition based on integrable functions. For simplicity, let's suppose that *f*(*t*)* *and *k*(*t*) are two real functions of a single variable with support in . The convolution of *f*(*t*) and *k*(*t*) (conventionally denoted as *f*(*t*) *** *k*(*t*)), which we are going to call a kernel, is defined as follows:

The expression may not be very easy to understand without a mathematical background, but it can become exceptionally simple with a few considerations. First of all, the integral sums all values of ; therefore, the convolution is a function of the remaining variable, *t*. The second fundamental element is a sort of dynamic property: the kernel is reversed () and transformed into a function of a new variable, . Without deep mathematical knowledge, it's possible to understand that this operation shifts the function along the (independent variable) axis...