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Questions tagged [wiener-algebra]

The Wiener algebra, usually denoted by $A(\mathbb{T})$, is the space of absolutely convergent Fourier series. Here $\mathbb{T}$ denotes the circle group. The norm of a function $f \in A(\mathbb{T})$ is $|f| = \sum\limits_{n = - \infty}^\infty |\hat{f}(n)|$.

The Wiener algebra, usually denoted by $A(\mathbb{T})$, is the space of absolutely convergent Fourier series. Here $\mathbb{T}$ denotes the circle group. The norm of a function $f \in A(\mathbb{T})$ is $\|f\| = \sum\limits_{n = - \infty}^\infty |\hat{f}(n)|$.

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Wiener Lemma for matrix valued functions

The ordinary Wiener lemma states that if $f(x):=\sum_{n \in \mathbb{Z}} a_n \exp(inx)$ ($\sum |a_n|<\infty$) and if $f(x)\neq 0$ everywhere, then $g:=1/f$ can also be written as $$g(x)= \sum b_n \exp(inx)$$ with $\sum_n |b_n|<\infty$ The proof of…
P.Jo
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Why is the Wiener Algebra closed under pointwise multiplication of functions?

I want to understand the proof given on the wikipedia page for the Wiener Algebra $A(\mathbb{T})$ (see here) that $A(\mathbb{T})$ is closed under multiplication. We have $$ A(\mathbb{T}) = \{f:\mathbb{T} \to \mathbb{R}: \|f\| < \infty \} $$ with…
Matt E.
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Problems about Wiener algebra(absolutely convergent fourier series)

The Wiener algebra $A(\mathbb T)$ is defined as the space of all functions defined on $\mathbb T$(the torus) such that its fourier series satisfies: $$\|f_n\|_{A(\mathbb T)}:=\sum\limits_{n}|\hat f(n)|<\infty.$$ An immediate consequence is stated:…
Christoff_ferland
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A problem on the Wiener Algebra

We say that $f\in C(\mathbb{T})$ is in the Wiener Algebra $A(\mathbb{T})$ if the series of Fourier coefficients $$\sum|\widehat{f}(n)|$$ is convergent (absolutely). It is verified that this sum defines a norm ,which we symbolize…
Pmorphy
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Bound of the norm of $f^{-1}$ in Wiener algebra

Setup. Wiener algebra $W$ is a set of all functions $f(\zeta) = \sum_{n\in \mathbb{Z}}c_n\zeta^n$ on the unit circle with $\|f\|_W = \sum_{n\in \mathbb{Z}}|c_n| < \infty$. The famous Wiener $1/f$ theorem states that $f$ is invertible in $W$ iff…
Pavel Gubkin
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Second moment of exponential of squared Wiener Process

I am a beginner in stochastics, studying it by myself using the book by Kurt Jacobs (Introduction to stochastic processes). In there is the following exercise: $x(t)= \mathrm{e}^{-b (W(t))^2}$ where $W(t)=\int^t_0 dW$ is a Wiener process. a)…
Nicouh
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Space of absolutely convergent Fourier series and density

It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : Approximating a continuous function by one with zero derivative). My question is as follows:…
M.G
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Norm in the space of absolutely convergent Fourier series

In the book Ensemble parfaits et séries trigonométriques, Kahane and Salem assert the following : $\big\|\exp\big(ing(x)\big)\big\|_{A(\mathbb{T})}$ is $O(|n|)$ where $g \in C^\infty(\mathbb{T})$ is real-valued and $n \in \mathbb{N}$. Here the…
M.G
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