Convergence in Schwartz Space of Fourier transforms?
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Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
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add a comment |
$begingroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
$endgroup$
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
$begingroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
$endgroup$
Let $f_n$ a sequence in Schwartz space. Suppose that $lim _{ nrightarrow infty }{ { left| { f }_{ n }-f right| }_{ alpha ,0 }=0 } $, where ${ left| f right| }_{ alpha ,0 }=underset { xin R }{ sup } { left| x right| }^{ alpha }left| fleft( x right) right|$. Is it true that its Fourier transforms converge in the same sense, in others words $lim _{ nrightarrow infty }{ { left| widehat { { f }_{ n }-f } right| }_{ alpha ,0 }= } 0$ or not?
convergence fourier-transform schwartz-space
convergence fourier-transform schwartz-space
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
edited Jan 22 at 12:27
Davide Giraudo
128k17156268
128k17156268
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
asked Jan 8 at 3:01
David LlerenaDavid Llerena
254
254
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
1
1
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
$begingroup$
No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51
add a comment |
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No as a bound for $|widehat{f}|_{alpha,0}$ for every $alpha$ gives a bound for $|f|_{0,m}$ for every $m$
$endgroup$
– reuns
Jan 8 at 4:51