Holder continuity of Fourier transform of measure












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I am trying to prove the following, say $mu$ is a non-negative Borel measure on $mathbb{R}$ with $mu(mathbb{R}) = 1$ and
$$ f(t) = widehat{mu}(t) = int_{mathbb{R}} e^{itx};dmu(x).$$
Assuming that
$$int_{mathbb{R}} |x|^{2+delta};dmu(x) < infty$$
Can we show that $fin mathrm{C}^{2,delta}(mathbb{R})$?






real-analysis measure-theory fourier-analysis fourier-transform







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    I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
    – reuns
    Nov 20 '18 at 0:42


















2














I am trying to prove the following, say $mu$ is a non-negative Borel measure on $mathbb{R}$ with $mu(mathbb{R}) = 1$ and
$$ f(t) = widehat{mu}(t) = int_{mathbb{R}} e^{itx};dmu(x).$$
Assuming that
$$int_{mathbb{R}} |x|^{2+delta};dmu(x) < infty$$
Can we show that $fin mathrm{C}^{2,delta}(mathbb{R})$?






real-analysis measure-theory fourier-analysis fourier-transform







share|cite|improve this question


















  • 2




    I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
    – reuns
    Nov 20 '18 at 0:42
















2












2








2







I am trying to prove the following, say $mu$ is a non-negative Borel measure on $mathbb{R}$ with $mu(mathbb{R}) = 1$ and
$$ f(t) = widehat{mu}(t) = int_{mathbb{R}} e^{itx};dmu(x).$$
Assuming that
$$int_{mathbb{R}} |x|^{2+delta};dmu(x) < infty$$
Can we show that $fin mathrm{C}^{2,delta}(mathbb{R})$?






real-analysis measure-theory fourier-analysis fourier-transform







share|cite|improve this question













I am trying to prove the following, say $mu$ is a non-negative Borel measure on $mathbb{R}$ with $mu(mathbb{R}) = 1$ and
$$ f(t) = widehat{mu}(t) = int_{mathbb{R}} e^{itx};dmu(x).$$
Assuming that
$$int_{mathbb{R}} |x|^{2+delta};dmu(x) < infty$$
Can we show that $fin mathrm{C}^{2,delta}(mathbb{R})$?






real-analysis measure-theory fourier-analysis fourier-transform




real-analysis measure-theory fourier-analysis fourier-transform






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 20 '18 at 0:09









Sean

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527513








  • 2




    I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
    – reuns
    Nov 20 '18 at 0:42
















  • 2




    I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
    – reuns
    Nov 20 '18 at 0:42










2




2




I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
– reuns
Nov 20 '18 at 0:42






I would start with $f''(t)-f''(t+c) = int_{|x| < c^{-r}}+int_{|x| > c^{-r}} x^2e^{itx} (e^{icx}-1) dmu(x)$ $= O( int |x|^2 c^{1-r} dmu(x))+O(int_{|x| > c^{-r}} |x|^2 dmu(x))$
– reuns
Nov 20 '18 at 0:42

















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