Function from weighted $L^p$ whith Fourier transform supported on finite measure set
Let $mu$ be a positive ($sigma$-finite) measure on $mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0neq fin L^p(mathbb{R},mu)$, for a fixed $pin[1,infty)$, such that its Fourier transform $hat f$ is supported on a set of finite measure,
$$
|mathbb{R}setminushat f^{-1}(0)|<infty.
$$
There are no further restrictions on $mu$.
Question: Does such a function $f$ exist?
Thank you.
Discussion: Since there are no restrictions on the growth of $mu$ at infinity, I cannot hope that $hat f$ is compactly supported, since Paley-Wiener functions cannot decay too fast. But finite measure support is a much weaker condition, I suppose.
One typical construction of functions supported on a set of finite measure is taking an infinite sum of shrunk and shifted copies of a single function of compact support. But since the union of supports of these copies has finite measure, such a system is not dense in $L^2$, and it is unclear how to make up a function with desired properties out of such building blocks.
If $mu$ is too "weak" then a function $fin L^p(mathbb{R},mu)$ may be very bad, so one may wonder what the Fourier transform of such an $f$ is. I am interested in such $f$ for which Fourier transform makes sense, for instance, $fin L^2(mathbb{R})$.
real-analysis functional-analysis fourier-analysis
asked 2 days ago
Bedovlat
596311
add a comment |
Let $mu$ be a positive ($sigma$-finite) measure on $mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0neq fin L^p(mathbb{R},mu)$, for a fixed $pin[1,infty)$, such that its Fourier transform $hat f$ is supported on a set of finite measure,
$$
|mathbb{R}setminushat f^{-1}(0)|<infty.
$$
There are no further restrictions on $mu$.
Question: Does such a function $f$ exist?
Thank you.
Discussion: Since there are no restrictions on the growth of $mu$ at infinity, I cannot hope that $hat f$ is compactly supported, since Paley-Wiener functions cannot decay too fast. But finite measure support is a much weaker condition, I suppose.
One typical construction of functions supported on a set of finite measure is taking an infinite sum of shrunk and shifted copies of a single function of compact support. But since the union of supports of these copies has finite measure, such a system is not dense in $L^2$, and it is unclear how to make up a function with desired properties out of such building blocks.
If $mu$ is too "weak" then a function $fin L^p(mathbb{R},mu)$ may be very bad, so one may wonder what the Fourier transform of such an $f$ is. I am interested in such $f$ for which Fourier transform makes sense, for instance, $fin L^2(mathbb{R})$.
real-analysis functional-analysis fourier-analysis
asked 2 days ago
Bedovlat
596311
add a comment |
Let $mu$ be a positive ($sigma$-finite) measure on $mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0neq fin L^p(mathbb{R},mu)$, for a fixed $pin[1,infty)$, such that its Fourier transform $hat f$ is supported on a set of finite measure,
$$
|mathbb{R}setminushat f^{-1}(0)|<infty.
$$
There are no further restrictions on $mu$.
Question: Does such a function $f$ exist?
Thank you.
Discussion: Since there are no restrictions on the growth of $mu$ at infinity, I cannot hope that $hat f$ is compactly supported, since Paley-Wiener functions cannot decay too fast. But finite measure support is a much weaker condition, I suppose.
One typical construction of functions supported on a set of finite measure is taking an infinite sum of shrunk and shifted copies of a single function of compact support. But since the union of supports of these copies has finite measure, such a system is not dense in $L^2$, and it is unclear how to make up a function with desired properties out of such building blocks.
If $mu$ is too "weak" then a function $fin L^p(mathbb{R},mu)$ may be very bad, so one may wonder what the Fourier transform of such an $f$ is. I am interested in such $f$ for which Fourier transform makes sense, for instance, $fin L^2(mathbb{R})$.
real-analysis functional-analysis fourier-analysis
asked 2 days ago
Bedovlat
596311
Let $mu$ be a positive ($sigma$-finite) measure on $mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0neq fin L^p(mathbb{R},mu)$, for a fixed $pin[1,infty)$, such that its Fourier transform $hat f$ is supported on a set of finite measure,
$$
|mathbb{R}setminushat f^{-1}(0)|<infty.
$$
There are no further restrictions on $mu$.
Question: Does such a function $f$ exist?
Thank you.
Discussion: Since there are no restrictions on the growth of $mu$ at infinity, I cannot hope that $hat f$ is compactly supported, since Paley-Wiener functions cannot decay too fast. But finite measure support is a much weaker condition, I suppose.
One typical construction of functions supported on a set of finite measure is taking an infinite sum of shrunk and shifted copies of a single function of compact support. But since the union of supports of these copies has finite measure, such a system is not dense in $L^2$, and it is unclear how to make up a function with desired properties out of such building blocks.
If $mu$ is too "weak" then a function $fin L^p(mathbb{R},mu)$ may be very bad, so one may wonder what the Fourier transform of such an $f$ is. I am interested in such $f$ for which Fourier transform makes sense, for instance, $fin L^2(mathbb{R})$.
real-analysis functional-analysis fourier-analysis
real-analysis functional-analysis fourier-analysis
asked 2 days ago
Bedovlat
596311
asked 2 days ago
Bedovlat
596311
edited 2 days ago
asked 2 days ago
Bedovlat
596311
asked 2 days ago
Bedovlat
596311
asked 2 days ago
Bedovlat
596311
596311
add a comment |
add a comment |
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