The Fourier transform of is always continuous and bounded on ; the transform of the convolution of two functions is the product of the transforms of each of the functions separately.
In the classical case of Fourier series, where (the additive group of integers) and (the circle group), the Fourier transform restricts to a unitary operator between the Hilbert spaces and and the Fourier coefficients are the numbers
for , where the functions form an orthonormal basis of . The Fourier transform is then viewed as the -series which in the -th place has and elsewhere . The Fourier transform replaces the operator of differentiation by the operator of multiplication by the series .
In general, if is a compact abelian group (whose Pontrjagin dual is discrete), one can normalize the invariant measure by and for . Then the Fourier transform restricts to a unitary operator from to .
A Fourier transform of a function on the real line is called its Fourier integral:
It is usually defined as a linear automorphism of the Schwarz space? ; there is also an appropriate extension to the space of distributions by where and . The Fourier transform and the inverse Fourier transform are continuous, mutually inverse operators . There is also a unitary operator on which when restricted to agrees with the Fourier transform.
The study of the Fourier transform is the main subject of Fourier analysis and, together with its generalizations like wavelet transform, of harmonic analysis. Regarding that, with an appropriate choice of functional spaces, the Fourier transform has an inverse, each function can be represented as a Fourier inverse of some function, which amounts to a decomposition into an integral over a Haar measure along some basis. Thus the function gets analysed into harmomics.
For noncommutative topological groups, instead of continuous characters one should consider irreducible unitary representations, which makes the subject much more difficult. There are also generalizations in noncommutative geometry, see quantum group Fourier transform.