Deriving an inequality for a set of functions that have equi-Lipschitz first derivatives.
$begingroup$
let $ F ={ F_t }_{t in mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' ={ F'_t }_{t in mathbb{N}}$ is equi-Lipschitz , i.e. there exist a real constant $L$ such that for all $x_1,x_2$
$$ |f(x_1) - f(x_2)| le L | x_1 - x_2 | $$
for all $f in F'$.
Then is it true that
$$F_{t+1} (x_1) - F_t(x_2) le F'_t(x_1) (x_2 -x_1) + frac{1}{2} L ( x_2 - x_1)^2 $$
for all $t$ ?
real-analysis continuity lipschitz-functions equicontinuity
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
$endgroup$
add a comment |
$begingroup$
let $ F ={ F_t }_{t in mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' ={ F'_t }_{t in mathbb{N}}$ is equi-Lipschitz , i.e. there exist a real constant $L$ such that for all $x_1,x_2$
$$ |f(x_1) - f(x_2)| le L | x_1 - x_2 | $$
for all $f in F'$.
Then is it true that
$$F_{t+1} (x_1) - F_t(x_2) le F'_t(x_1) (x_2 -x_1) + frac{1}{2} L ( x_2 - x_1)^2 $$
for all $t$ ?
real-analysis continuity lipschitz-functions equicontinuity
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
$endgroup$
add a comment |
$begingroup$
let $ F ={ F_t }_{t in mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' ={ F'_t }_{t in mathbb{N}}$ is equi-Lipschitz , i.e. there exist a real constant $L$ such that for all $x_1,x_2$
$$ |f(x_1) - f(x_2)| le L | x_1 - x_2 | $$
for all $f in F'$.
Then is it true that
$$F_{t+1} (x_1) - F_t(x_2) le F'_t(x_1) (x_2 -x_1) + frac{1}{2} L ( x_2 - x_1)^2 $$
for all $t$ ?
real-analysis continuity lipschitz-functions equicontinuity
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
$endgroup$
let $ F ={ F_t }_{t in mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' ={ F'_t }_{t in mathbb{N}}$ is equi-Lipschitz , i.e. there exist a real constant $L$ such that for all $x_1,x_2$
$$ |f(x_1) - f(x_2)| le L | x_1 - x_2 | $$
for all $f in F'$.
Then is it true that
$$F_{t+1} (x_1) - F_t(x_2) le F'_t(x_1) (x_2 -x_1) + frac{1}{2} L ( x_2 - x_1)^2 $$
for all $t$ ?
real-analysis continuity lipschitz-functions equicontinuity
real-analysis continuity lipschitz-functions equicontinuity
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
asked Jan 8 at 15:32
MonoliteMonolite
1,5412925
1,5412925
add a comment |
add a comment |
1 Answer
1
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$begingroup$
No. Let $F_t(x)=t$ for all $xinBbb R$. Then $F'_tequiv0$ fot all $tinBbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $tinBbb N$ and all $x_i,x_2inBbb R$.
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
add a comment |
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1 Answer
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active
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1 Answer
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active
oldest
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active
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votes
$begingroup$
No. Let $F_t(x)=t$ for all $xinBbb R$. Then $F'_tequiv0$ fot all $tinBbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $tinBbb N$ and all $x_i,x_2inBbb R$.
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
add a comment |
$begingroup$
No. Let $F_t(x)=t$ for all $xinBbb R$. Then $F'_tequiv0$ fot all $tinBbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $tinBbb N$ and all $x_i,x_2inBbb R$.
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
add a comment |
$begingroup$
No. Let $F_t(x)=t$ for all $xinBbb R$. Then $F'_tequiv0$ fot all $tinBbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $tinBbb N$ and all $x_i,x_2inBbb R$.
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
No. Let $F_t(x)=t$ for all $xinBbb R$. Then $F'_tequiv0$ fot all $tinBbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $tinBbb N$ and all $x_i,x_2inBbb R$.
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
answered Jan 8 at 15:42
Julián AguirreJulián Aguirre
68.1k24094
68.1k24094
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
add a comment |
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Thank you! Obviously! how dumb of me, would some hypothesis on the way the functions in the sequence change make it true? or at-least approximately true?
$endgroup$
– Monolite
Jan 8 at 15:44
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
Some condition on the differences $F_{t+1}-F_t$ is needed.
$endgroup$
– Julián Aguirre
Jan 8 at 15:46
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
$begingroup$
I'll write another question, thanks again!
$endgroup$
– Monolite
Jan 8 at 15:51
add a comment |
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