Fourier transform is first studied as a transform on functions on a whole space, such as , with basic results concerning inversion, convolution, Plancherel's theorem, and the decomposition of functions into frequencies. A restriction problem asks the question of whether the Fourier transform of a function can be meaningfully restricted to a lower-dimensional set in frequency space, such as a sphere, cone, or paraboloid.

If , then is continuous, so its restriction to a smooth surface is well defined. The problem becomes nontrivial for