Beurling algebra

In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling (1949), usually it is an algebra of periodic functions with Fourier series

$f(x)=\sum a_n{e}^{inx}$

Example We may consider the algebra of those functions f where the majorants

$c_k=\underset{|n|\ge k}{\sup }|a_n|$

of the Fourier coefficients a_n are summable. In other words

$\sum _{k\ge 0}{c}_k<\mathrm{\infty }.$

Example We may consider a weight function w on $\mathbb{\mathbb{Z}}$ such that

$w(m+n)\le w(m)w(n),w(0)=1$

in which case $A_w(\mathbb{𝕋})=\{f:f(t)=\sum _n{a}_n{e}^{int},\Vert f{\Vert }_w=\sum _n|a_n|w(n)<\mathrm{\infty }\}(\sim {\ell }_w^1(\mathbb{\mathbb{Z}}))$ is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• Hazewinkel, Michiel, ed. (2001), "Beurling algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Beurling, Arne (1949), "On the spectral synthesis of bounded functions", Acta Math. 81 (1): 225–238, doi:10.1007/BF02395018