Beurling algebra
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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
- [math]\displaystyle{ f(x)=\sum a_ne^{inx} }[/math]
Example We may consider the algebra of those functions f where the majorants
- [math]\displaystyle{ c_k=\sup_{|n|\ge k} |a_n| }[/math]
of the Fourier coefficients an are summable. In other words
- [math]\displaystyle{ \sum_{k\ge 0} c_k\lt \infty. }[/math]
Example We may consider a weight function w on [math]\displaystyle{ \mathbb{Z} }[/math] such that
- [math]\displaystyle{ w(m+n)\leq w(m)w(n),\quad w(0)=1 }[/math]
in which case [math]\displaystyle{ A_w(\mathbb{T}) =\{f:f(t)=\sum_na_ne^{int},\,\|f\|_w=\sum_n|a_n|w(n)\lt \infty\} \,(\sim\ell^1_w(\mathbb{Z})) }[/math] 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
Original source: https://en.wikipedia.org/wiki/Beurling algebra.
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