Convergence Uniform / Holder continuous
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Let $f: R->R$ be a $2L-periodic$ function. Suppose there are 'The positive constant and $α$ $∈$ $(0, 1)$
such that
$| f (x) - f (y) | ≤ A(| x - y |^α + | x - y |) $∀ x,y $ and ∈ [-L, L].$
Show that $Sn$ converges uniformly to $f$ in any compact set of $R$
I was thinking of proving that $ f $ is continuous holder,
soon satisfies the Dini test, then $ Sn $ converges to $ f $.
But how can I ensure that convergence is uniform?
Should I use the fact that f is bounded variation?
real-analysis fourier-analysis
asked Nov 17 at 22:28
justlearningmath
175
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up vote
0
down vote
favorite
Let $f: R->R$ be a $2L-periodic$ function. Suppose there are 'The positive constant and $α$ $∈$ $(0, 1)$
such that
$| f (x) - f (y) | ≤ A(| x - y |^α + | x - y |) $∀ x,y $ and ∈ [-L, L].$
Show that $Sn$ converges uniformly to $f$ in any compact set of $R$
I was thinking of proving that $ f $ is continuous holder,
soon satisfies the Dini test, then $ Sn $ converges to $ f $.
But how can I ensure that convergence is uniform?
Should I use the fact that f is bounded variation?
real-analysis fourier-analysis
asked Nov 17 at 22:28
justlearningmath
175
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f: R->R$ be a $2L-periodic$ function. Suppose there are 'The positive constant and $α$ $∈$ $(0, 1)$
such that
$| f (x) - f (y) | ≤ A(| x - y |^α + | x - y |) $∀ x,y $ and ∈ [-L, L].$
Show that $Sn$ converges uniformly to $f$ in any compact set of $R$
I was thinking of proving that $ f $ is continuous holder,
soon satisfies the Dini test, then $ Sn $ converges to $ f $.
But how can I ensure that convergence is uniform?
Should I use the fact that f is bounded variation?
real-analysis fourier-analysis
asked Nov 17 at 22:28
justlearningmath
175
Let $f: R->R$ be a $2L-periodic$ function. Suppose there are 'The positive constant and $α$ $∈$ $(0, 1)$
such that
$| f (x) - f (y) | ≤ A(| x - y |^α + | x - y |) $∀ x,y $ and ∈ [-L, L].$
Show that $Sn$ converges uniformly to $f$ in any compact set of $R$
I was thinking of proving that $ f $ is continuous holder,
soon satisfies the Dini test, then $ Sn $ converges to $ f $.
But how can I ensure that convergence is uniform?
Should I use the fact that f is bounded variation?
real-analysis fourier-analysis
real-analysis fourier-analysis
asked Nov 17 at 22:28
justlearningmath
175
asked Nov 17 at 22:28
justlearningmath
175
asked Nov 17 at 22:28
justlearningmath
175
asked Nov 17 at 22:28
justlearningmath
175
asked Nov 17 at 22:28
justlearningmath
175
175
add a comment |
add a comment |
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