In mathematics, the Wiener algebra, named after

Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...

and usually denoted by , is the space of

absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

. Here denotes the

circle group In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...

Banach algebra structure

The norm of a function is given by :

\, f\, =\sum_^\infty , \hat(n), ,\,

where :

\hat(n)= \frac\int_^\pi f(t)e^ \, dt

is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed, :

\begin
f(t)g(t) & = \sum_ \hat(m)e^\,\cdot\,\sum_ \hat(n)e^ \\
& = \sum_ \hat(m)\hat(n)e^ \\
& = \sum_ \left\e^
,\qquad f,g\in A(\mathbb);
\end

therefore :

\, f g\,  
= \sum_ \left,  \sum_ \hat(n-m)\hat(m) \ 
\leq \sum_ , \hat(m),  \sum_n , \hat(n),  = \, f\,  \, \, g\, .\,

Thus the Wiener algebra is a commutative unitary Banach algebra. Also, is isomorphic to the Banach algebra , with the isomorphism given by the Fourier transform.

Properties

The sum of an absolutely convergent Fourier series is continuous, so :

A(\mathbb)\subset C(\mathbb)

where is the ring of continuous functions on the unit circle. On the other hand an

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...

, together with the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

and Parseval's formula, shows that :

C^1(\mathbb)\subset A(\mathbb).\,

More generally, :

\mathrm_\alpha(\mathbb)\subset A(\mathbb)\subset C(\mathbb)

for

\alpha>1/2

(see ).

Wiener's 1/''f'' theorem

proved that if has absolutely convergent Fourier series and is never zero, then its reciprocal also has an absolutely convergent Fourier series. Many other proofs have appeared since then, including an elementary one by . used the theory of Banach algebras that he developed to show that the maximal ideals of are of the form :

M_x = \left\, \quad x \in \mathbb~,

which is equivalent to Wiener's theorem.

Notes

References

* * * * * * * {{SpectralTheory Banach algebras Fourier series

Banach algebra structure

Properties

Wiener's 1/''f'' theorem

See also

Notes

References