Properties of Finite DifferencesI have seen this statement in different variants, but could not find a proof:If the n-th order differences of equally spaced data are non-zero and constant then the data can be represented by a polynomial.I interpret this statement more formally as follows:Let f : Z → Z be any function such that the finite differences D r , defined by D r + 1 ( x ) = D r ( x + 1 ) − D r ( x ) for r ≥ 0 (where D 0 := f by convention)are such that D r is constant and non-zero for any r ≥ r 0 . Then f is a polynomial in x (with Q coefficients) and degree r 0 .Question: Is this correct as I stated it? How does one prove that such f must be a polynomial?

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Properties of Finite DifferencesI have seen this statement in different variants, but could not find a proof:If the n-th order differences of equally spaced data are non-zero and constant then the data can be represented by a polynomial.I interpret this statement more formally as follows:Let   f  :      Z    →      Z  be any function such that the finite differences       D    r  , defined by      D          r      +      1        (  x  )  =      D    r    (  x  +  1  )  −      D    r    (  x  ) for   r  ≥  0 (where       D    0    :=  f by convention)are such that       D    r   is constant and non-zero for any   r  ≥      r    0  . Then f is a polynomial in x (with Q coefficients) and degree       r    0  .Question: Is this correct as I stated it? How does one prove that such f must be a polynomial?

trajeronls · Accepted Answer

Every polynomial of degree d can be expressed in the basis given by the Falling Factorials up to falling index d.That is                          p        (        x        )        =                  a                                  d                                            x                                  d                          +                  a                                  d            −            1                                            x                                  d            −            1                          +                ⋯                +                  a                                  0                                            x                                  0                          =                                  =                  b                                  d                                            x                                                d              _                                      +                  b                                  d            −            1                                            x                                                                d                −                1                            _                                      +                ⋯                +                  b                                  0                                            x                                                0              _                                          Then, from the properties of the falling factorial, it is easy to demonstrate that      Δ    n    p  (  0  )  =      ∑    k                      (                  −          1                )                    n        −        k                    (                                    n                                                k                              )            p    (    k    )    =      {                                        0                                              d              &amp;lt;              n                                                                          d              !                              b                                                    d                                            =              d              !                              a                                                    d                                                                                        n              =              d                                                                          n              !                              b                                                    n                                                                                        n              ≤              d                                              And viceversa, given      Δ    d    p  (  x  )  =  d  !      b                d          ⇒        Δ          d      +      1        p  (  x  )  =  0the Newton series assures us that  p  (  a  +  x  )  =  p  (  a  )  +  x    Δ  p  (  a  )  +  ⋯  +                              x                                                                                d                            _                                                  d        !                    Δ                d        p  (  a  )So we can conclude thatif we are given a sequence of m+1 points equally spaced by h      {                  x        0            ,              x        0            +      h      ,              x        0            +      2      h      ,      …      ,              x        0            +      m      h        }  and the corresponding sequence of values taken by a function f(x)            {              f        (                  x          0                +        k        h        )            }              k      =      0        h  we can put  f      (                  x        0            +      k      h        )    =  f      (          h              (                                                                              x                  0                                            h                                +          k                )              )    =  g      (    k    )  denoting the differences as      {                            Δ          g                      (            k            )                    =          g                      (                          k              +              1                        )                    −          g                      (            k            )                                                            Δ            n                    g                      (            k            )                    =          Δ                      (                                          Δ                                  n                  −                  1                                            g                              (                k                )                                      )                    =                                      =                      ∑                                          (                                  0                  ≤                                )                            j                              (                                  ≤                  n                                )                                                                                        (                                  −                  1                                )                                            n                −                j                                                    (                                                                    n                                                                                        j                                                              )                        g                          (                              k                +                j                            )                                                      |                                        0              ≤              n              +              k              ≤              m                                                          we have that the sequence in n of   (  −  1      )    n        Δ    n    g      (    k    )   is the Binomial Transform of the sequence of the values g(k+n)            {                                    (                          −              1                        )                    n                          Δ          n                g                  (          k          )                    }              n      =      0              m      −      k                                  ⟷                    B        i        n        o        m        i        a        l                T        r        a        n        s        f        o        r        m                        {              g                  (                      k            +            n                    )                    }              n      =      0              m      −      k      and since the Binomial Transform is self-inverse, we get  g      (          k      +      n        )    =      ∑                  (                  0          ≤                )            j              (                  ≤          n                )                        (                                    n                                                j                              )              Δ      j        g          (      k      )        =      ∑                  (                  0          ≤                )            j              (                  ≤          n                )                                                      Δ            j                    g                      (            k            )                                    j          !                                    (                              (                          n              +              k                        )                    −          k                )                                                      j                                _                    which is precisely the Newton series.given n+1 points   {  k  ,  k  +  1  ,  …  ,  k  +  n  } and the respective g(k+j)the Newton series is the unique polynomial of max degree n which interpolates them.if the differences of order n&amp;lt;m are all equal to each other (at changing k)      Δ    n    g      (    k    )    =  c  ≠  0        |                0      ≤      ∀      k      ≤      m      −      n          then all the differences of order higher than n will be null      Δ          n      +      q        g      (    k    )    =      Δ    q        (                  Δ        n            g              (        k        )              )    =      Δ    q    c  =  0        |                ∀      k          and the relevant Newton series will be the unique polynomial of degree n which interpolates all the m+1 points

Properties of Finite Differences I have seen this statement in different variants, but could not fi

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