Fourier Transform On Finite Abelian Groups
Since the irreducible representations of finite abelian groups are all of degree 1 and hence equal to the irreducible characters of the group, Fourier analysis on finite abelian groups is significantly simplified. For instance, the Fourier transform yields a scalar- and not matrix-valued function.
Furthermore, the irreducible characters of a group may be put in one-to-one correspondence with the elements of the group.
Therefore, we may define the Fourier transform for finite abelian groups as
Note that the right-hand side is simply for the inner product on the vector space of functions from to defined by
The inverse Fourier transform is then given by
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply where 0 is the group identity and is the Kronecker delta.
Read more about this topic: Fourier Transform On Finite Groups
Famous quotes containing the words transform, finite and/or groups:
“He had said that everything possessed
The power to transform itself, or else,
And what meant more, to be transformed.”
—Wallace Stevens (1879–1955)
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)