Question

$f$ absolutely continuous and $f'\in L^3[0,1]$, which values of $\alpha $ does $\lim_{x\rightarrow 0^+} x^{-\alpha}f(x)=0$?

score 53 · Answer 1 · 2026-04-14 22:26:03

53

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$f$ absolutely continuous and $f'\in L^3[0,1]$, which values of $\alpha $ does $\lim_{x\rightarrow 0^+} x^{-\alpha}f(x)=0$?

Published on 14 Apr 2026 - 22:26

score 69 · Answer 2 · 2026-04-13 16:38:57

69

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Does there exist a function $f\in L^1(\mathbb{R})$ such that $\int_{-\infty}^{+\infty}e^{-|y|}f(x-y)dy=e^{-|x|}$ for all $x\in \mathbb{R}$

Published on 13 Apr 2026 - 16:38

score 140 · Answer 3 · 2026-04-10 23:04:23

140

Views

$L^p$ compactness for product of two sequences of functions

Published on 10 Apr 2026 - 23:04

score 503 · Answer 4 · 2026-04-17 11:55:51

503

Views

prove that $f_n\in L^1_\text{loc}(\mathbb{R})$

Published on 17 Apr 2026 - 11:55

score 93 · Answer 5 · 2026-04-12 00:17:43

93

Views

Convergence in $p$th mean does not imply convergence in mean.

Published on 12 Apr 2026 - 0:17

score 122 · Answer 6 · 2026-03-25 23:50:54

122

Views

closed subspaces in $\ell^p$

Published on 25 Mar 2026 - 23:50

score 64 · Answer 7 · 2026-04-14 12:40:48

64

Views

Show that $X \in L^p$

Published on 14 Apr 2026 - 12:40

score 79 · Answer 8 · 2026-04-13 17:35:26

79

Views

Completeness of $\mathcal{l}^1$?

Published on 13 Apr 2026 - 17:35

score 1.6k · Answer 9 · 2026-04-15 21:40:54

1.6k

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Show that the $L^1$ and $L^2$ norms are not equivalent on the set of continuous functions from $[0,1]$ to $\mathbb{R}$

Published on 15 Apr 2026 - 21:40

score 80 · Answer 10 · 2026-04-23 00:33:51

80

Views

For $1<p<2, $ Fourier transform is not onto $L^p(\Bbb T) \to \ell^q(\Bbb Z)$ where $\frac{1}{p}+\frac{1}{q}=1$

Published on 23 Apr 2026 - 0:33

score 169 · Answer 11 · 2026-04-15 09:58:15

169

Views

Example for a $p$-integrable function which vanishes locally a.e. but not globally a.e.

Published on 15 Apr 2026 - 9:58

score 258 · Answer 12 · 2026-04-11 19:23:43

258

Views

Bartle's Elements of Integration Exercise 6.J

Published on 11 Apr 2026 - 19:23

score 225 · Answer 13 · 2026-04-09 07:50:32

225

Views

Definition essential supremum

Published on 09 Apr 2026 - 7:50

score 134 · Answer 14 · 2026-04-15 07:36:58

134

Views

Solution verification : Truncation of $L^p$ functions

Published on 15 Apr 2026 - 7:36

score 86 · Answer 15 · 2026-04-24 09:23:27

86

Views

Lebesgue Limit Stochastic

Published on 24 Apr 2026 - 9:23

List Question

$f$ absolutely continuous and $f'\in L^3[0,1]$, which values of $\alpha $ does $\lim_{x\rightarrow 0^+} x^{-\alpha}f(x)=0$?

Does there exist a function $f\in L^1(\mathbb{R})$ such that $\int_{-\infty}^{+\infty}e^{-|y|}f(x-y)dy=e^{-|x|}$ for all $x\in \mathbb{R}$

$L^p$ compactness for product of two sequences of functions

prove that $f_n\in L^1_\text{loc}(\mathbb{R})$

Convergence in $p$th mean does not imply convergence in mean.

closed subspaces in $\ell^p$

Show that $X \in L^p$

Completeness of $\mathcal{l}^1$?

Show that the $L^1$ and $L^2$ norms are not equivalent on the set of continuous functions from $[0,1]$ to $\mathbb{R}$

For $1<p<2, $ Fourier transform is not onto $L^p(\Bbb T) \to \ell^q(\Bbb Z)$ where $\frac{1}{p}+\frac{1}{q}=1$

Example for a $p$-integrable function which vanishes locally a.e. but not globally a.e.

Bartle's Elements of Integration Exercise 6.J

Definition essential supremum

Solution verification : Truncation of $L^p$ functions

Lebesgue Limit Stochastic

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