This article about polynomial interpolation claims that (it is known that) every rational function may be represented in barycentric form: r ( x ) = ∑ j = 0 N w j x − x j y j ∑ j = 0 N w j x − x j What about the rational function r ( x , y ) = x y x 2 + y 2 ? But even if I accept that we are talking only about univariate functions, I would still like to know a bit more about the cited statement:1. Is this statement strictly true for R , i.e. for any rational function r ( x ) = p ( x ) q ( x ) with p , q ∈ R [ x ] (polynomials with real coefficients) there are w j , x j , y j ∈ R representing r(x)?2. Is this statement strictly true for C , i.e. for any rational function r ( x ) = p ( x ) q ( x ) with p , q ∈ C [ x ] (polynomials with complex coefficients) there are w j , x j , y j ∈ C representing r(x)?3. For r ( x ) = p ( x ) q ( x ) and N = deg ⁡ p + deg ⁡ q, is the set of x j ∈ C for which appropriate w j , y j ∈ C exist open and dense in C N + 1 ?4. Where can I learn more about this topic, since when is this known, ... ?

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This article about polynomial interpolation claims that (it is known that) every rational function may be represented in barycentric form:  r  (  x  )  =                    ∑                  j          =          0                N                              w          j                          x          −                      x            j                                      y        j                            ∑                  j          =          0                N                              w          j                          x          −                      x            j                              What about the rational function   r  (  x  ,  y  )  =            x      y                      x        2            +              y        2            ? But even if I accept that we are talking only about univariate functions, I would still like to know a bit more about the cited statement:1. Is this statement strictly true for       R  , i.e. for any rational function   r  (  x  )  =            p      (      x      )              q      (      x      )       with   p  ,  q  ∈      R    [  x  ] (polynomials with real coefficients) there are       w    j    ,      x    j    ,      y    j    ∈      R   representing r(x)?2. Is this statement strictly true for       C  , i.e. for any rational function   r  (  x  )  =            p      (      x      )              q      (      x      )       with   p  ,  q  ∈      C    [  x  ] (polynomials with complex coefficients) there are       w    j    ,      x    j    ,      y    j    ∈      C   representing r(x)?3. For   r  (  x  )  =            p      (      x      )              q      (      x      )       and   N  =  deg  ⁡  p  +  deg  ⁡  q, is the set of       x    j    ∈      C   for which appropriate       w    j    ,      y    j    ∈      C   exist open and dense in             C              N      +      1      ?4. Where can I learn more about this topic, since when is this known, ... ?

Nina Luna · Accepted Answer

The claim is that any rational function   r  (  x  )  =            p      (      x      )              q      (      x      )       with   max  (  deg  ⁡  p  ,  deg  ⁡  q  )  ≤  n for n+1  distinct interpolation points       x    j   with   q  (      x    j    )  ≠  0 can be written in barycentric form  r  (  x  )  =                    ∑                  j          =          0                n                              u          j                          x          −                      x            j                                      r        j                            ∑                  j          =          0                n                              u          j                          x          −                      x            j                              for       r    j    :=  r  (      x    j    ) and suitable weights       u    j  . The derivation is actually quite simple:Let   l  (  x  )  =  (  x  −      x    0    )  (  x  −      x    1    )  …  (  x  −      x    n    ) and       w    j    =                                    x          −                      x            j                                    l          (          x          )                    |              x      =              x        j            . The barycentric Lagrange formula represents the denominator as   q  (  x  )  =  l  (  x  )      ∑          j      =      0        n              w      j              x      −              x        j                  q    j  . For             p      ~        (  x  )  :=  l  (  x  )      ∑          j      =      0        n              w      j              x      −              x        j                  q    j        r    j   we have                              p          ~                    (      x      )              q      (      x      )        =                    ∑                  j          =          0                n                              w          j                          x          −                      x            j                                      q        j                    r        j                            ∑                  j          =          0                n                              w          j                          x          −                      x            j                                      q        j              =                    ∑                  j          =          0                n                              u          j                          x          −                      x            j                                      r        j                            ∑                  j          =          0                n                              u          j                          x          −                      x            j                              with       u    j    :=      w    j        q    j  , and             p      ~        (  x  )  =  p  (  x  ) follows from   deg  ⁡  p  ≤  n and       q    j    =  q  (      x    j    )  ≠  0. We have   p  (      x    j    )  =      q    j        r    j  , because       q    j    =  q  (      x    j    )  ≠  0. So             p      ~        (  x  ) coincides with p(x)  at n+1 points, and both are polynomials of degree less than n+1, so they are identical.

This article about polynomial interpolation claims that (it is known that) every rational function m

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