Supplementary Problems

We assume that the sequence of complex numbers $z_{1}, z_{2}, \ldots , z_{k}, \ldots$ has the following three properties:

1. The series

   $$\sum _{k=1}^{\infty } z_{k}, \qquad \sum _{k=1}^{\infty } z_{k}^{2}, \qquad \ldots , \qquad \sum _{k=1}^{\infty } z_{k}^{n}$$

   are convergent.

2. $\left|z_{1}\right| \geq \left|z_{2}\right| \geq \cdots \geq \left|z_{k}\right| \geq \cdots$.

3. There exist two real numbers $\alpha$ and $\beta$ such that

   $$0 < \beta -\alpha < \frac{2\pi n}{n+1}, \qquad \alpha \leq \arg z_{k} \leq \beta , \qquad \text{for } k=1,2,3,\ldots .$$

Then $\left|z_{1}\right|^{n}+\left|z_{2}\right|^{n}+\cdots +\left|z_{k}\right|^{n}+\cdots$ is convergent.

The conclusion holds under a wider assumption: The series listed in (1) need not be convergent; it is sufficient to assume that their partial sums are bounded. Yet the conclusion becomes invalid if we drop assumption (2) or if we require less than the first part of (3), namely that

The functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z)$; $g_{1}(z), g_{2}(z), \ldots , g_{n}(z)$ are regular analytic in a region of the $z$-plane where

If $f_{1}(z), f_{2}(z), \ldots , f_{n}(z)$ are linearly independent, then so also are $g_{1}(z), g_{2}(z), \ldots , g_{n}(z)$, and

for $\nu =1,2,\ldots ,n$; the constants $c_{\nu \lambda }$ form a unitary matrix.

We consider the approximation of the entire function $g(z)$ by polynomials. We define the integer

(termed the degree of approximation of the function $g(z)$ on the circle $\left|z\right|=r$ to accuracy $\alpha$) as the degree of the polynomial $P(z)$ of the lowest possible degree for which

($r>0$, $\alpha >1$). Then

where we keep $r$ fixed when computing the first limit and $\alpha$ fixed when computing the second limit; $\rho$ is the order of $g(z)$.

Let $g(z)$ be an entire function whose growth does not exceed the minimum type of order $1$; that is, $g(z)e^{-\varepsilon \left|z\right|}$ remains bounded in the whole $z$-plane provided that $\varepsilon$ is a fixed positive number. Assume that the sequence

is bounded. Then $g(z)$ reduces to a constant.

We consider square matrices with $n$ rows and $n$ columns, and let capital letters denote such matrices. Let

be an entire function and $Z$ a matrix. Then the series

converges to a well-defined matrix.

Given the matrix $A$, when does there exist a matrix $Z$ such that

Let $f(z)$ be a single-valued analytic function which is regular everywhere including the point $z=\infty$, except at the point $z=1$. In a neighborhood of the point $z=0$ it has the expansion

and in a neighborhood of the point $z=\infty$ the expansion

If there exists a constant $k$ such that both $a_{n}n^{-k}$ and $b_{n}n^{-k}$ are bounded ($n=1,2,3,\ldots$), then $f(z)$ is a rational function.

Let the numbers $a, b, m_{1}, m_{2}, m_{3}, \ldots$ be real, and satisfy

Furthermore, let $f(x)$ be continuous in the interval $[a,b]$. Then it follows from

for $n=1,2,3,\ldots$, that $f(x)$ vanishes identically. (A fraction of the whole system of Fourier functions is closed over the corresponding fraction of the whole interval.)