Rational interpolation ($q(x_i)/r(x_i)=f_i$)

Rational interpolation seeks a rational function $q/r$ with prescribed bounds on the polynomial degrees of its numerator and denominator and prescribed values at distinct interpolation nodes. A rational interpolant must have a well-defined value at each interpolation node. With a specified numerator-denominator representation, this requires $r(x_i)\ne 0$ for every $i$. A normalization such as a monic polynomial denominator removes the common scaling freedom but does not guarantee existence or uniqueness. Multiplying by a zero denominator can create spurious solutions to the rational interpolation linearization.

For $0<k\le 1$, seek a rational function $r(x)=ax/(1+bx)$ that has $r(0)=0$, $r(1)=1$ and $r^{\prime }(1)=k$, matching the value and first derivative of $x^k$ at one. The last two conditions give $a=1+b$ and $a/(1+b{)}^2=k$, henceFor $x\in [0,1]$ its denominator is positive, so it has no pole there. The usual degree-$(1,1)$ Padé approximant about one is instead $[(1-k)+(1+k)x]/[(1+k)+(1-k)x]$, matching the next Taylor series coefficient while generally failing the zero-endpoint condition.

The limiting case $k=0$ needs care: for each fixed $x>0$, the constrained expression tends to one, while its value at $x=0$ remains zero for every $k>0$. Cancelling $x/x$ gives the constant extension one only after abandoning the condition $r(0)=0$. Thus the three original constraints cannot all hold in this family at $k=0$.