Is it possible to construct a trigonometric series convergent in $(0,1)$ while divergent in $(2,3)$?
Here by a trigonometric series, I mean
$$ f(x)=sum_{n=1}^infty a_n e^{ i b_n x }, $$
where $a_n$, $b_n$ can be arbitrary complex numbers.
Two Questions:
Q1. Is it possible to make such a function $f$ convergent in $(0,1)$ but divergent in $(2,3)$?
Q2. As a related question, is it possible to make $f$ differentiable in $(0,1)$ but not differentiable in $(2,3)$?
fourier-analysis fourier-series
edited Nov 28 at 23:21
Mason
1,8581529
asked Nov 28 at 23:10
pie
735
add a comment |
Here by a trigonometric series, I mean
$$ f(x)=sum_{n=1}^infty a_n e^{ i b_n x }, $$
where $a_n$, $b_n$ can be arbitrary complex numbers.
Two Questions:
Q1. Is it possible to make such a function $f$ convergent in $(0,1)$ but divergent in $(2,3)$?
Q2. As a related question, is it possible to make $f$ differentiable in $(0,1)$ but not differentiable in $(2,3)$?
fourier-analysis fourier-series
edited Nov 28 at 23:21
Mason
1,8581529
asked Nov 28 at 23:10
pie
735
Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Yes, pointwise.
– pie
Nov 29 at 11:02
add a comment |
Here by a trigonometric series, I mean
$$ f(x)=sum_{n=1}^infty a_n e^{ i b_n x }, $$
where $a_n$, $b_n$ can be arbitrary complex numbers.
Two Questions:
Q1. Is it possible to make such a function $f$ convergent in $(0,1)$ but divergent in $(2,3)$?
Q2. As a related question, is it possible to make $f$ differentiable in $(0,1)$ but not differentiable in $(2,3)$?
fourier-analysis fourier-series
edited Nov 28 at 23:21
Mason
1,8581529
asked Nov 28 at 23:10
pie
735
Here by a trigonometric series, I mean
$$ f(x)=sum_{n=1}^infty a_n e^{ i b_n x }, $$
where $a_n$, $b_n$ can be arbitrary complex numbers.
Two Questions:
Q1. Is it possible to make such a function $f$ convergent in $(0,1)$ but divergent in $(2,3)$?
Q2. As a related question, is it possible to make $f$ differentiable in $(0,1)$ but not differentiable in $(2,3)$?
fourier-analysis fourier-series
fourier-analysis fourier-series
edited Nov 28 at 23:21
Mason
1,8581529
asked Nov 28 at 23:10
pie
735
edited Nov 28 at 23:21
Mason
1,8581529
asked Nov 28 at 23:10
pie
735
edited Nov 28 at 23:21
Mason
1,8581529
edited Nov 28 at 23:21
Mason
1,8581529
edited Nov 28 at 23:21
Mason
1,8581529
1,8581529
asked Nov 28 at 23:10
pie
735
asked Nov 28 at 23:10
pie
735
asked Nov 28 at 23:10
pie
735
735
Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Yes, pointwise.
– pie
Nov 29 at 11:02
add a comment |
Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Yes, pointwise.
– pie
Nov 29 at 11:02
Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Yes, pointwise.
– pie
Nov 29 at 11:02
Yes, pointwise.
– pie
Nov 29 at 11:02
add a comment |
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Converges in what sense? Pointwise?
– Dunham
Nov 29 at 1:19
Yes, pointwise.
– pie
Nov 29 at 11:02