Necessary and sufficient conditions for the existence of the Newton Series of a function $f: mathbb{N}...
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I’m wondering if a function $f: mathbb{N} longrightarrow R$ can be represented as a Newton series given that all its forward differences exist.
The first thing I searched up was a result in complex analysis called Carlson’s Theorem which helps to show when a function in $mathbb{C}$ is identical to its Newton series.
But the functions I’m concerned with only has domain $mathbb{N}$. With my naive understanding of Taylor series’s, I know that if a function is infinitely differentiable at a point, then it is identical to its Taylor series at that point.
Can something analogous be true for Newton series’s if I restrict functions to domain $mathbb{N}$ instead of $mathbb{C}$ without any extra conditions such as those in Carlson’s Theorem?
real-analysis discrete-calculus newton-series
asked Nov 18 at 13:05
zetapenguin
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I’m wondering if a function $f: mathbb{N} longrightarrow R$ can be represented as a Newton series given that all its forward differences exist.
The first thing I searched up was a result in complex analysis called Carlson’s Theorem which helps to show when a function in $mathbb{C}$ is identical to its Newton series.
But the functions I’m concerned with only has domain $mathbb{N}$. With my naive understanding of Taylor series’s, I know that if a function is infinitely differentiable at a point, then it is identical to its Taylor series at that point.
Can something analogous be true for Newton series’s if I restrict functions to domain $mathbb{N}$ instead of $mathbb{C}$ without any extra conditions such as those in Carlson’s Theorem?
real-analysis discrete-calculus newton-series
asked Nov 18 at 13:05
zetapenguin
64
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I’m wondering if a function $f: mathbb{N} longrightarrow R$ can be represented as a Newton series given that all its forward differences exist.
The first thing I searched up was a result in complex analysis called Carlson’s Theorem which helps to show when a function in $mathbb{C}$ is identical to its Newton series.
But the functions I’m concerned with only has domain $mathbb{N}$. With my naive understanding of Taylor series’s, I know that if a function is infinitely differentiable at a point, then it is identical to its Taylor series at that point.
Can something analogous be true for Newton series’s if I restrict functions to domain $mathbb{N}$ instead of $mathbb{C}$ without any extra conditions such as those in Carlson’s Theorem?
real-analysis discrete-calculus newton-series
asked Nov 18 at 13:05
zetapenguin
64
I’m wondering if a function $f: mathbb{N} longrightarrow R$ can be represented as a Newton series given that all its forward differences exist.
The first thing I searched up was a result in complex analysis called Carlson’s Theorem which helps to show when a function in $mathbb{C}$ is identical to its Newton series.
But the functions I’m concerned with only has domain $mathbb{N}$. With my naive understanding of Taylor series’s, I know that if a function is infinitely differentiable at a point, then it is identical to its Taylor series at that point.
Can something analogous be true for Newton series’s if I restrict functions to domain $mathbb{N}$ instead of $mathbb{C}$ without any extra conditions such as those in Carlson’s Theorem?
real-analysis discrete-calculus newton-series
real-analysis discrete-calculus newton-series
asked Nov 18 at 13:05
zetapenguin
64
asked Nov 18 at 13:05
zetapenguin
64
edited Nov 18 at 13:49
asked Nov 18 at 13:05
zetapenguin
64
asked Nov 18 at 13:05
zetapenguin
64
asked Nov 18 at 13:05
zetapenguin
64
64
add a comment |
add a comment |
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