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Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} v_{1,0}&v_{2,0}&\dots&v_{n,0}\\ v_{1,1}&v_{2,1}&\dots&v_{n,1}\\ \vdots\\ v_{1,N-1}&v_{2,N-1}&\dots&v_{n,N-1} \end{bmatrix}$$ where $v_{j,k}=\begin{bmatrix}x_j^k,&kx_j^{k-1},&\dots&k(k-1)\times\dots\times (k-l_j+1) x_j^{k-l_j+1}\end{bmatrix}$. Let $\|\cdot\|$ denote the row sum matrix norm. In some applications (e.g. interpolation, signal processing) one would like to estimate the quantity $\|V^{-1}\|$.

Gautschi [1] has shown that for $l_1=\dots=l_n=2$ one has $$\|V^{-1}\| \leq \max_{1\leq \lambda\leq n} \beta_{\lambda} \prod_{\nu=1,\nu\neq\lambda}^n \biggl(\frac{1+|x_{\lambda}|}{|x_{\nu}-x_{\lambda}|}\biggr)^2$$ where $\beta_{\lambda}=\max\biggl(1+|x_{\lambda}|,1+2(1+|x_{\lambda}|)\sum_{\nu\neq\lambda}{1\over |x_\nu-x_\lambda|}\biggr)$.

I am interested in a somewhat cruder estimates, as follows: if $|x_j|\leq 1$ and $|x_i-x_j|\geq \delta$, then for the above case we have $$\|V^{-1}\| \leq C n 2^N\delta^{-N+1}\qquad (*)$$ for some absolute constant $C$.

Is it true that something like $(*)$ holds for the general configuration $\{l_1,\dots,l_n\}$?

EDIT: using [2], this seems to boil down to the following. Let $$h_j(x)=\prod_{i \neq j}(x-x_i)^{-l_i}.$$ For $t=0,1,\dots,l_j$ evaluate $h_j^{(t)}(x_j).$

[1] W.Gautschi, "On Inverses of Vandermonde and Confluent Vandermonde matrices II", Numerische Mathematik 5, 425-430, 1963.

[2] R.Schapelle, "The Inverse of the Confluent Vandermonde Matrix", IEEE Trans. on Automatic Control, October 1972, pp.724-725.

2 cleanup of formula

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} v_{1,0}&v_{2,0}&\dots&v_{n,0}\\ v_{1,1}&v_{2,1}&\dots&v_{n,1}\\ \vdots\\ v_{1,N-1}&v_{2,N-1}&\dots&v_{n,N-1} \end{bmatrix}$$ where $v_{j,k}=\begin{bmatrix}x_j^k,&kx_j^{k-1},&\dots&k(k-1)\times\dots\times (k-l_j+1) x_j^{k-l_j+1}\end{bmatrix}$. Let $\|\cdot\|$ denote the row sum matrix norm. In some applications (e.g. interpolation, signal processing) one would like to estimate the quantity $\|V^{-1}\|$.

Gautschi [1] has shown that for $l_1=\dots=l_n=2$ one has $$\|V^{-1}\| \leq \max_{1\leq \lambda\leq n} \beta_{\lambda} \prod_{\nu=1,\nu\neq\lambda}^n \biggl(\frac{(1+|x_{\lambda}|)}{|x_{\nu}-x_{\lambda}|}\biggr)^2$$ biggl(\frac{1+|x_{\lambda}|}{|x_{\nu}-x_{\lambda}|}\biggr)^2$$where \beta_{\lambda}=\max\biggl(1+|x_{\lambda}|,1+2(1+|x_{\lambda}|)\sum_{\nu\neq\lambda}{1\over |x_\nu-x_\lambda|}\biggr). I am interested in a somewhat cruder estimates, as follows: if |x_j|\leq 1 and |x_i-x_j|\geq \delta, then for the above case we have$$\|V^{-1}\| \leq C n 2^N\delta^{-N+1}\qquad (*) for some absolute constant $C$.

Is it true that something like $(*)$ holds for the general configuration $\{l_1,\dots,l_n\}$?

[1] W.Gautschi, "On Inverses of Vandermonde and Confluent Vandermonde matrices II", Numerische Mathematik 5, 425-430, 1963.

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