Birkhoff interpolation

Remarks

The conditions form a square linear system in the $n + 1$ coefficients of $p$, so poisedness means that its determinant is nonzero. If in each row of $E$ the ones form an initial block $e_{i0} = \cdots = e_{i,k_i - 1} = 1$, the problem is Hermite interpolation, which is always poised. If moreover each row has a single one in column $0$, it is polynomial interpolation.

Example: nodes $x_1 = 0$, $x_2 = 1$, $n = 1$, conditions $p(0) = a$ and $p'(1) = b$. For $p = c_0 + c_1 x$ these give $c_0 = a$ and $c_1 = b$, so the problem is poised.

Non-example: nodes $-1, 0, 1$, $n = 2$, conditions $p(-1) = a$, $p'(0) = b$, $p(1) = c$. For $p = c_0 + c_1 x + c_2 x^2$ the system has the matrix with rows $(1, -1, 1)$, $(0, 1, 0)$, $(1, 1, 1)$, whose determinant is $0$. For instance $p = x^2 - 1$ is a nonzero solution with zero data, so the problem is not poised.