The latex code here is doing all sorts of odd things... :( ... anyway,(adsbygoogle = window.adsbygoogle || []).push({});

The convolution algebra is [tex]l_1(\mathbb{Z},\mathbb{C})[/tex], the set of functions [tex]f:\mathbb{Z}\rightarrow\mathbb{C}[/tex] which satisfy

[tex]||f||:=\sum_{n=-\infty}^{\infty}|f(n)|<\infty[/tex]

with pointwise addition and scalar multiplication, and multiplication of functions defined by

[tex]f*g(n)=\sum_{m=-\infty}^{\infty}f(n-m)g(m)[/tex]

(this is a commutative Banach algebra).

For [tex]z\in\mathbb{T}[/tex], the unit circle in the complex plane, the functional [tex]\psi_z:l_1(\mathbb{Z},\mathbb{C})\rightarrow\mathbb{C}[/tex] is defined by

[tex]\psi_z(f)=\sum_{n=-\infty}^{infty}f(n)z^n[/tex]

.[tex]\psi_z[/tex] is a non-zero homomorphism (in fact the set of all of these is the set of all non-zero complex homomorphisms).

For a function f in the algebra, the gelfand transform is

[tex]\hat{f}(\psi_z)=\psi_z(f)=\sum_{n=-\infty}^{\infty}f(n)z^n[/tex]

The example I'm trying to understand shows how to find the inverse of a particular function f.

Part of the working says that "hat" is injective, so [at this stage the latex code is being absolutely ridiculous - priting something I had in a previous sentence that I subsequently deleted, and the actual thing I want is nowhere to be found :( ... I want to write that you can interchange the order of "inversing" and "hatting" f]. This I don't understand... it's probably quite simple, but something's not clicking for me unfortunately.

Thanks for any help.

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# Convolution algebra - help understanding a worked example

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