the convolution theorem

Two operations that look completely different turn out to be two sides of the same coin once you pass to frequencies. In physical space, blending one function through another — smearing, averaging, filtering — is done by a convolution. In frequency space, that messy blending is nothing but ordinary multiplication. The convolution theorem is the bridge that says so, and it is the engine behind almost every closed-form PDE solution gotten by transforms.

First, what a convolution is: the convolution of f and g, written (f * g)(x) = integral from -infinity to infinity of f(y) g(x - y) dy, slides g across f and accumulates the overlap — it is exactly how a smoothing or averaging kernel acts on data. The theorem then states that the Fourier transform of a convolution is the plain product of the transforms: the transform of (f * g) equals f-hat(xi) times g-hat(xi). Conversely, multiplying two functions in physical space corresponds to convolving their transforms. So a hard integral collapses into a pointwise product the instant you look at it in frequency.

This is exactly why the heat kernel appears. Solving u_t = k u_xx by transform gives u-hat(xi, t) = f-hat(xi) e^(-k*xi^2 t) — a product of two transforms. By the convolution theorem the answer in physical space is u(x, t) = (f * G)(x, t), the initial data convolved with the inverse transform G of e^(-k*xi^2 t), which is the Gaussian heat kernel. The same trick builds the solution of any constant-coefficient linear PDE as a convolution of the data with the equation's fundamental solution. Honest caveat: the integral defining f * g must converge, so you still need f and g to decay sensibly for the clean statement to hold.