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?










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    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?










    share|cite|improve this question


























      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?










      share|cite|improve this question















      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






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      share|cite|improve this question













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      edited Nov 18 at 13:49

























      asked Nov 18 at 13:05









      zetapenguin

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