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I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $$ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & -k_{1} & 1\\ -k_{3} & -k_{2} & -k_{1} & 1\\ \vdots & & & \ddots & \ddots \end{array}\right] $$ And the coefficients $0<k_i<1$ are monotone decreasing with $i$ and convergent ($\sum_{i=1}^{\infty}k_{i}=c_0<1$). Moreover, we assume $\lim_{i\rightarrow\infty}\frac{k_{i+1}}{k_i}=s_0$ i.e., $\{k_i\}$ is geometric series when $i$ is large.

Now, let $T=Q^{-1}$ is the inverse of this Toeplitz matrix. I read other papers and know $T$ is also lower triangular Teoplitz matrix. Assume the coefficients are $t_i$, and denote $$ T=Q^{-1}=\left[\begin{array}{ccccc} 1\\ t_{1} & 1\\ t_{2} & t_{1} & 1\\ t_{3} & t_{2} & t_{1} & 1\\ \vdots & & & & \ddots \end{array}\right] $$

I want to know whether $\{t_i\}$ are convergent, and moreover, will $\frac{t_{i+1}}{t_i}$ converge to a constant when $i$ is large enough ($i\rightarrow\infty$)? I tried some examples, and the results showed that both of them are convergent to a constant. But I don't know how to prove it.

I hope to discuss with you. Thanks in advance.

[Updated]

Considering @mike's idea, now I am thinking whether we can express the question as the coefficient ratio for a power series.

As $Q=L^{0}+\sum_{i=1}^{\infty}\left(-k_{i}L^{i}\right)$, where $L$ is the shift matrix in which $L_{ij}=\delta_{i,j+1}$ in infinite-dimensional ($L^0=I$). And $T$ can be written as $T=L^{0}+\sum_{i=1}^{\infty}\left(t_{i}L^{i}\right)$.

So the question is whether ${t_i}$ is convergent and whether $\lim_{i\rightarrow\infty}\frac{t_{i+1}}{t_i}$ is a constant ($z_0$). If yes, what's the expression of $z_0$.

Is this related to the power-series? I don't have much knowledge about it...

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You know Weiner's Thm on the spectrum of $\mathbb A(\mathbb T)$ (absolutely summable) which can be looked up in Katznelson's book ? If you had the circulant matrix it would apply directly, and tell you that what you are interested in is whether $1-\sum k_ie^{2 \pi i t}$ ever vanishes. –  mike Aug 29 '12 at 12:49
    
Well, I will think about this way...Thanks @mike. But how to check whether $\frac{t_{i+1}}{t_i}$ is convergent, which is more important to me actually. –  Chang Aug 29 '12 at 17:08
    
The relationship between a triangular Toeplitz matrix and formal power series is discussed in Henrici's "Applied and Computational Complex Analysis", vol. 1, Ch. 1. (The book calls it a semicirculant but it is the same thing.) –  AcidFlask Mar 20 '13 at 17:12
    
Thank you @AcidFlask. I will check it later. Thanks for the help. –  Chang Mar 20 '13 at 21:15
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1 Answer

up vote 2 down vote accepted

The key here is that $q=1-\sum_{i>0}k_ix^i$ is the power series around $0$ with radius of convergence $\frac1{s_0}$, and that $1+\sum_{i>0}t_ix^i$ is the series for its reciprocal $\frac1q$ (which series can be written $\sum_{n\geq 0}(\sum_{i>0}k_ix^i)^n$, so in particular all $t_i$ are positive). I can't come up right now with a setting in which the statement "$Q$ is the matrix of multiplication by $q$ on the basis of monomials, and $T$ the matrix of multiplication by $\frac1q$" makes sense formally, but morally that's what is going on.

The question of whether the $t_i$ converge is negative in general. Just take $k_1=a>1$ and $k_i=0$ for $i>1$, then $t_i=a^i$ diverges. (OK, you don't allow me to take $k_i=0$, but taking $k_i>0$ will only make the $t_i$ larger.) In fact the informaion about the $k_i$ tells you about the singularities of $f$, but for the $t_i$ you need infomation about the zeroes of $f$. Since by continuity you know that $f$ is nonzero on some neighborhood of $0$, you know that $\frac1q$ has positive radius of convergence and therefore $\sqrt[n]{t_n}$ is bounded; this does not imply though that $\frac{t_{i+1}}{t_i}$ converges. Maybe you can do something with the condition that $\frac{k_{i+1}}{k_i}$ convergerges, but I can't see just how.

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Hi @Marc, nice to see you again. And I really appreciate your help. Actually I am re-thinking the problem recently, and it now is the problem <math.stackexchange.com/questions/188872/…; like this. What I am really concerned is the limit of the ratio ($\frac{t_{i+1}}{t_i}$). You will find that the question in that link is based on this question. And I found that $\lim_{i\rightarrow\infty}\frac{k_{i+1}}{k_i}$ is a constant. But I cannot prove the convergent right now. Anyway, thank you so much~~ –  Chang Aug 31 '12 at 15:19
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