Tagged Questions

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Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
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Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
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The behavior of Fourier transform near the origin

I'm attacking a homework problem, which I have reduced to the following: Let Schwartz function $f \in \mathcal{S}^1(\mathbb{R})$ be nonnegative and $\|f\|_{L^1} = 1$. Assume further that ...
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How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)=$The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu$; and ...
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How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
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Find the Hardy-Littlewood maximal function of $\chi_{[-1,1]}$ on $\Bbb R$

Find the Hardy-Littlewood maximal function $Mf$ of the $\Bbb R^\Bbb R$ function $f=\chi_{[-1,1]}$. How do we find $Mf(x)$ for $|x| > 1$? I see that it should decrease like $1/x$, but I can't find ...
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Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
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Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
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Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$

I would to link the simple function and probabilistic approach for the calculation of the Fourier transform of the Cantor function. Let $f:[0,1] \to [0,1]$ be the Cantor function. In the simple ...
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Probabilistic calulation of the Fourier transform of the Cantor function

This is on the same theme as in this post, where the Fourier transform was derived using simple function. Let $f:[0,1] \to [0,1]$ be the Cantor function. Then $f$ is the cumulative distribution of ...
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A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
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Does convergence of Fourier transforms imply convergence of measures?

Let $\{\sigma_n\}$ be a sequence of measures on the complex unit circle $\mathbb{T}$ and let $\sigma$ also be such a measure. Suppose that $\hat{\sigma_n}(k) \rightarrow \hat{\sigma}(k)$ as ...
I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing ...