Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum

Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $?

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down vote

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Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?

I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.

Is that relation true? I am struggling to prove this.

Please help me!

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

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up vote
3
down vote

favorite

Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?

I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.

Is that relation true? I am struggling to prove this.

Please help me!

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

add a commentÂ |Â

up vote
3
down vote

favorite

Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?

I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.

Is that relation true? I am struggling to prove this.

Please help me!

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

Suppose $fin L_2(mathbb R)$ and $f$ is a continuous function on $mathbb R$. Is $displaystylesum_kinmathbbZ|f(k)|^2<infty $ ?

I am trying to prove the relation $displaystylesum_kinmathbbZ|f(k)|^2<int_mathbb R|f(x)|^2dx$.

Is that relation true? I am struggling to prove this.

Please help me!

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

asked Aug 19 at 4:39

S. Pitchai Murugan

287111

add a commentÂ |Â

1 Answer
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Let $t(x) = max(0, 1-|x|)$, note that $t(0) = 1$ and $|t|^2 = 2 over 3$.

Note that $| x mapsto t(k^2x) |^2 = 1over k^2 2 over 3$.

Let $f(x) = sum_k neq 0 t(k^2(x-k))$, note that $|f|^2 le 2 over 3 sum_k 1 over k^2$
and $f(k) = 1$ (for $k neq 1$), hence $sum_k |f(k)|^2 $ is unbounded.

(I ignored $k=0$ so the support of each of the constituent functions in the sum has disjoint supports. Not necessary, but makes the relationship
$g(f(x)) = sum_k neq 0 g(t(k^2(x-k)))$ true for
any function $g$.)

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

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1 Answer
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1 Answer
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active

oldest

votes

up vote
6
down vote

Let $t(x) = max(0, 1-|x|)$, note that $t(0) = 1$ and $|t|^2 = 2 over 3$.

Note that $| x mapsto t(k^2x) |^2 = 1over k^2 2 over 3$.

Let $f(x) = sum_k neq 0 t(k^2(x-k))$, note that $|f|^2 le 2 over 3 sum_k 1 over k^2$
and $f(k) = 1$ (for $k neq 1$), hence $sum_k |f(k)|^2 $ is unbounded.

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

123k557156

add a commentÂ |Â

up vote
6
down vote

Let $t(x) = max(0, 1-|x|)$, note that $t(0) = 1$ and $|t|^2 = 2 over 3$.

Note that $| x mapsto t(k^2x) |^2 = 1over k^2 2 over 3$.

Let $f(x) = sum_k neq 0 t(k^2(x-k))$, note that $|f|^2 le 2 over 3 sum_k 1 over k^2$
and $f(k) = 1$ (for $k neq 1$), hence $sum_k |f(k)|^2 $ is unbounded.

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

123k557156

add a commentÂ |Â

up vote
6
down vote

Let $t(x) = max(0, 1-|x|)$, note that $t(0) = 1$ and $|t|^2 = 2 over 3$.

Note that $| x mapsto t(k^2x) |^2 = 1over k^2 2 over 3$.

Let $f(x) = sum_k neq 0 t(k^2(x-k))$, note that $|f|^2 le 2 over 3 sum_k 1 over k^2$
and $f(k) = 1$ (for $k neq 1$), hence $sum_k |f(k)|^2 $ is unbounded.

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

123k557156

Let $t(x) = max(0, 1-|x|)$, note that $t(0) = 1$ and $|t|^2 = 2 over 3$.

Note that $| x mapsto t(k^2x) |^2 = 1over k^2 2 over 3$.

Let $f(x) = sum_k neq 0 t(k^2(x-k))$, note that $|f|^2 le 2 over 3 sum_k 1 over k^2$
and $f(k) = 1$ (for $k neq 1$), hence $sum_k |f(k)|^2 $ is unbounded.

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

123k557156

edited Aug 19 at 5:02

answered Aug 19 at 4:57

copper.hat

123k557156

answered Aug 19 at 4:57

copper.hat

123k557156

answered Aug 19 at 4:57

copper.hat

123k557156

add a commentÂ |Â

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