Problem on Riemann-Stieltjes Integration and function approximation
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Let $alpha$ be continuous and increasing function on [a,b]. Given $f$ $in$ ${R_alpha }$[a,b] and $epsilon$> 0;
Then, Prove that there exist
(i) a step function $h$ on [a,b] with ${||h||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - h|dalpha < varepsilon }$$
(ii)a continuous function $g$ on [a,b] with ${||g||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - g|dalpha < varepsilon }$$
(Def:$space$${R_alpha }$[a,b]=space of all Riemann-Stieltjes integrable functions w.r.t $alpha$ on the interval [a,b])
functional-analysis functions elementary-set-theory riemann-integration approximation-theory
asked Nov 16 at 16:24
HindShah
74
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up vote
0
down vote
favorite
Let $alpha$ be continuous and increasing function on [a,b]. Given $f$ $in$ ${R_alpha }$[a,b] and $epsilon$> 0;
Then, Prove that there exist
(i) a step function $h$ on [a,b] with ${||h||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - h|dalpha < varepsilon }$$
(ii)a continuous function $g$ on [a,b] with ${||g||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - g|dalpha < varepsilon }$$
(Def:$space$${R_alpha }$[a,b]=space of all Riemann-Stieltjes integrable functions w.r.t $alpha$ on the interval [a,b])
functional-analysis functions elementary-set-theory riemann-integration approximation-theory
asked Nov 16 at 16:24
HindShah
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $alpha$ be continuous and increasing function on [a,b]. Given $f$ $in$ ${R_alpha }$[a,b] and $epsilon$> 0;
Then, Prove that there exist
(i) a step function $h$ on [a,b] with ${||h||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - h|dalpha < varepsilon }$$
(ii)a continuous function $g$ on [a,b] with ${||g||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - g|dalpha < varepsilon }$$
(Def:$space$${R_alpha }$[a,b]=space of all Riemann-Stieltjes integrable functions w.r.t $alpha$ on the interval [a,b])
functional-analysis functions elementary-set-theory riemann-integration approximation-theory
asked Nov 16 at 16:24
HindShah
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $alpha$ be continuous and increasing function on [a,b]. Given $f$ $in$ ${R_alpha }$[a,b] and $epsilon$> 0;
Then, Prove that there exist
(i) a step function $h$ on [a,b] with ${||h||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - h|dalpha < varepsilon }$$
(ii)a continuous function $g$ on [a,b] with ${||g||_infty}$$leq$${||f||_infty}$ such that
$$intlimits_a^b {|f - g|dalpha < varepsilon }$$
(Def:$space$${R_alpha }$[a,b]=space of all Riemann-Stieltjes integrable functions w.r.t $alpha$ on the interval [a,b])
functional-analysis functions elementary-set-theory riemann-integration approximation-theory
functional-analysis functions elementary-set-theory riemann-integration approximation-theory
asked Nov 16 at 16:24
HindShah
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 16 at 16:24
HindShah
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 45 mins ago
asked Nov 16 at 16:24
HindShah
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 16 at 16:24
HindShah
74
asked Nov 16 at 16:24
HindShah
74
74
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
HindShah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02
add a comment |
For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02
For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02
For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02
add a comment |
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HindShah is a new contributor. Be nice, and check out our Code of Conduct.
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For (i), try functions of the form $h(x) = frac{1}{N} lfloor N f(x) rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral.
– Connor Harris
Nov 16 at 17:02