Find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not...
up vote
6
down vote
favorite
I am trying to find a function $f: mathbb{R}^+ to mathbb{R}^+$ that fullfils the following conditions
$$f in mathcal{C}^1(mathbb{R}^+,mathbb{R}^+)$$
$$int_{mathbb{R}^+} f in mathbb{R}^+$$
$$int_{mathbb{R}^+} mid f' mid in mathbb{R}$$
$$f text{is not $frac{1}{2}$-Hölder}$$
I've tried functions with smooth spikes but I am unable to express this function as combinations of usual functions.
Moreover, I know from this post that if $f'^2$ is integrable then $f$ is necessarily $frac{1}{2}-$Hölder.
Thank you.
Note: all the integrals are taken in the Riemann sense.
calculus real-analysis integration indefinite-integrals sobolev-spaces
edited 15 hours ago
RGS
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
6
down vote
favorite
I am trying to find a function $f: mathbb{R}^+ to mathbb{R}^+$ that fullfils the following conditions
$$f in mathcal{C}^1(mathbb{R}^+,mathbb{R}^+)$$
$$int_{mathbb{R}^+} f in mathbb{R}^+$$
$$int_{mathbb{R}^+} mid f' mid in mathbb{R}$$
$$f text{is not $frac{1}{2}$-Hölder}$$
I've tried functions with smooth spikes but I am unable to express this function as combinations of usual functions.
Moreover, I know from this post that if $f'^2$ is integrable then $f$ is necessarily $frac{1}{2}-$Hölder.
Thank you.
Note: all the integrals are taken in the Riemann sense.
calculus real-analysis integration indefinite-integrals sobolev-spaces
edited 15 hours ago
RGS
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
what does your second integral mean?
– zhw.
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
I am trying to find a function $f: mathbb{R}^+ to mathbb{R}^+$ that fullfils the following conditions
$$f in mathcal{C}^1(mathbb{R}^+,mathbb{R}^+)$$
$$int_{mathbb{R}^+} f in mathbb{R}^+$$
$$int_{mathbb{R}^+} mid f' mid in mathbb{R}$$
$$f text{is not $frac{1}{2}$-Hölder}$$
I've tried functions with smooth spikes but I am unable to express this function as combinations of usual functions.
Moreover, I know from this post that if $f'^2$ is integrable then $f$ is necessarily $frac{1}{2}-$Hölder.
Thank you.
Note: all the integrals are taken in the Riemann sense.
calculus real-analysis integration indefinite-integrals sobolev-spaces
edited 15 hours ago
RGS
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am trying to find a function $f: mathbb{R}^+ to mathbb{R}^+$ that fullfils the following conditions
$$f in mathcal{C}^1(mathbb{R}^+,mathbb{R}^+)$$
$$int_{mathbb{R}^+} f in mathbb{R}^+$$
$$int_{mathbb{R}^+} mid f' mid in mathbb{R}$$
$$f text{is not $frac{1}{2}$-Hölder}$$
I've tried functions with smooth spikes but I am unable to express this function as combinations of usual functions.
Moreover, I know from this post that if $f'^2$ is integrable then $f$ is necessarily $frac{1}{2}-$Hölder.
Thank you.
Note: all the integrals are taken in the Riemann sense.
calculus real-analysis integration indefinite-integrals sobolev-spaces
calculus real-analysis integration indefinite-integrals sobolev-spaces
edited 15 hours ago
RGS
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 15 hours ago
RGS
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 15 hours ago
RGS
8,85111129
edited 15 hours ago
RGS
8,85111129
edited 15 hours ago
RGS
8,85111129
8,85111129
asked 2 days ago
IBPsilly
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
IBPsilly
514
asked 2 days ago
IBPsilly
514
514
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
IBPsilly is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
what does your second integral mean?
– zhw.
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago
add a comment |
what does your second integral mean?
– zhw.
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago
what does your second integral mean?
– zhw.
2 days ago
what does your second integral mean?
– zhw.
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
IBPsilly is a new contributor. Be nice, and check out our Code of Conduct.
IBPsilly is a new contributor. Be nice, and check out our Code of Conduct.
IBPsilly is a new contributor. Be nice, and check out our Code of Conduct.
IBPsilly is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2994090%2ffind-f-such-that-f-is-absolutely-integrable-f-is-absolutely-integrable%23new-answer', 'question_page');
}
);
Post as a guest
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
what does your second integral mean?
– zhw.
2 days ago
@zhw I am sorry, this was a typo. It should be better now.
– IBPsilly
2 days ago