Math Problems Vault

Problem 7.55

We term the determinant

\begin{vmatrix} f_{1}(x) & f_{1}'(x) & f_{1}''(x) & \ldots & f_{1}^{(n-1)}(x) \\ f_{2}(x) & f_{2}'(x) & f_{2}''(x) & \ldots & f_{2}^{(n-1)}(x) \\ f_{3}(x) & f_{3}'(x) & f_{3}''(x) & \ldots & f_{3}^{(n-1)}(x) \\ \multicolumn {5}{c}{\dotfill } \\ f_{n}(x) & f_{n}'(x) & f_{n}''(x) & \ldots & f_{n}^{(n-1)}(x) \end{vmatrix} = W[f_{1}(x), f_{2}(x), \ldots , f_{n}(x)]

the Wronskian determinant of the system of functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ .

If $c_{\lambda \mu }$ are constants, then we have

W(c_{11} f_{1} + c_{12} f_{2} + \cdots + c_{1n} f_{n}, \; c_{21} f_{1} + c_{22} f_{2} + \cdots + c_{2n} f_{n}, \; \ldots , \; c_{n1} f_{1} + c_{n2} f_{2} + \cdots + c_{nn} f_{n}) = \left|c_{\lambda \mu }\right|_{1}^{n} \cdot W(f_{1}, f_{2}, \ldots , f_{n}).

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Problem 7.56

W[f_{1}(\varphi (x)), f_{2}(\varphi (x)), \ldots , f_{n}(\varphi (x))] = \varphi '(x)^{\frac{n(n-1)}{2}} W[f_{1}(y), f_{2}(y), \ldots , f_{n}(y)],

setting $y = \varphi (x)$ on the right-hand side.

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Problem 7.57

W(\varphi f_{1}, \varphi f_{2}, \ldots , \varphi f_{n}) = \varphi ^{n} W(f_{1}, f_{2}, \ldots , f_{n}).

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Problem 7.58

\frac{1}{f_{1}^{n}} W(f_{1}, f_{2}, \ldots , f_{n}) = W\left[ \left( \frac{f_{2}}{f_{1}} \right)', \left( \frac{f_{3}}{f_{1}} \right)', \ldots , \left( \frac{f_{n}}{f_{1}} \right)' \right].

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Problem 7.59

\frac{d}{dx} \frac{W(f_{1}, \ldots , f_{n-2}, f_{n})}{W(f_{1}, \ldots , f_{n-2}, f_{n-1})} = \frac{W(f_{1}, \ldots , f_{n-2}) W(f_{1}, \ldots , f_{n-2}, f_{n-1}, f_{n})}{[W(f_{1}, \ldots , f_{n-2}, f_{n-1})]^{2}}.

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Problem 7.60

If $W(f_{1}, f_{2}, \ldots , f_{n-1}, f_{n})$ vanishes at every point and $W(f_{1}, f_{2}, \ldots , f_{n-1})$ at no point of the interval $a, b$ , then there exist $n-1$ constants $c_{1}, c_{2}, \ldots , c_{n-1}$ such that in the whole interval $a, b$

f_{n}(x) = c_{1} f_{1}(x) + c_{2} f_{2}(x) + \cdots + c_{n-1} f_{n-1}(x).

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Problem 7.61

If $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ are $n$ linearly independent solutions of the homogeneous linear $n$ th order differential equation

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y = 0,

then we have for an arbitrary function $y$

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y = \frac{W(f_{1}, f_{2}, \ldots , f_{n}, y)}{W(f_{1}, f_{2}, \ldots , f_{n})}.

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Problem 7.62

(continued). Set

1 = W_{0}, \qquad f_{1} = W_{1}, \qquad W(f_{1}, f_{2}) = W_{2}, \qquad \ldots , \qquad W(f_{1}, f_{2}, \ldots , f_{n}) = W_{n}.

Then we have for an arbitrary function $y$

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y

\equiv \frac{W_{n}}{W_{n-1}} \cdot \frac{d}{dx} \frac{W_{n-1}^{2}}{W_{n-2} W_{n}} \cdots \frac{d}{dx} \frac{W_{2}^{2}}{W_{1} W_{3}} \cdot \frac{d}{dx} \frac{W_{1}^{2}}{W_{0} W_{2}} \cdot \frac{d}{dx} \frac{y}{W_{1}}.

This representation of the linear differential expression, in terms of $n$ independent solutions of the homogeneous differential equation, is analogous to the factorization of a polynomial of degree $n$ in terms of its $n$ zeros.

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Problem 7.63

Let $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ be continuous real functions, defined in the interval $a, b$ . The determinant

\left|\int _{a}^{b} f_{\lambda }(x) f_{\mu }(x) \, dx\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It vanishes if and only if between the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ there exists a homogeneous linear relation with constant coefficients, not all $0$ .

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Problem 7.64

(continued). The determinant

\begin{vmatrix} \displaystyle \int _{a}^{b} (f_{2}^{2}+f_{3}^{2}+\cdots +f_{n}^{2}) \, dx & -\displaystyle \int _{a}^{b} f_{1} f_{2} \, dx & \ldots & -\displaystyle \int _{a}^{b} f_{1} f_{n} \, dx \\ ]2ex] -\displaystyle \int _{a}^{b} f_{2} f_{1} \, dx & \displaystyle \int _{a}^{b} (f_{1}^{2}+f_{3}^{2}+\cdots +f_{n}^{2}) \, dx & \ldots & -\displaystyle \int _{a}^{b} f_{2} f_{n} \, dx \\[2ex] \multicolumn {4}{c}{\dotfill } \\ ]1ex] -\displaystyle \int _{a}^{b} f_{n} f_{1} \, dx & -\displaystyle \int _{a}^{b} f_{n} f_{2} \, dx & \ldots & \displaystyle \int _{a}^{b} (f_{1}^{2}+f_{2}^{2}+\cdots +f_{n-1}^{2}) \, dx \end{vmatrix} \[ is never negative. It vanishes if and only if the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ differ from one another only by constant factors, or, in other words, if they are all multiples of one of them. \]

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Problem 7.65

(continued). The determinant

\left|e^{\int _{a}^{b} f_{\lambda }(x) f_{\mu }(x) \, dx}\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It vanishes if and only if among the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ there are two identically equal.

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Problem 7.66

A theorem mentioned in passing by Gauss in his Theoria combinationis observationum [Werke, Vol. 4, p. 12, cf. the last two lines of Art. 11] may be generalized as follows:

Let $f(x)$ be defined for $x > 0$ , non-increasing there, $f(x) \geq 0$ , but not identically $=0$ . Moreover let the real numbers $a_{1}, a_{2}, \ldots , a_{n}$ be distinct. Assuming the existence of all the integrals that occur, the determinant

\left|(a_{\lambda }+a_{\mu }+1) \int _{0}^{\infty } x^{a_{\lambda }+a_{\mu }} f(x) \, dx\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It is $=0$ if and only if $f(x)$ is a piecewise constant function with not more than $n-1$ jump discontinuities.

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Problem 7.67

Let the function $f(\vartheta )$ be properly integrable in the interval $0 \leq \vartheta < 2\pi$ . Let its Fourier series (Section 4 of Part VI) be

f(\vartheta ) \sim a_{0} + 2 \sum _{n=1}^{\infty } (a_{n} \cos n\vartheta + b_{n} \sin n\vartheta ).

The Hermite forms

T_{n}(f) = \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

formed from the constants $c_{0} = a_{0}$ , $c_{n} = a_{n} + ib_{n}$ , $c_{-n} = \bar{c}_{n}$ , $n = 1, 2, 3, \ldots$ , are called the Toeplitz forms associated with $f(\vartheta )$ . (Cf. Problem 43).

Form the Toeplitz forms that belong to the following functions:

f(\vartheta ) = c = \text{const.}, \qquad f(\vartheta ) = a_{0} + 2(a_{1} \cos \vartheta + b_{1} \sin \vartheta ),

f(\vartheta ) = \frac{1-r^{2}}{1-2r\cos \vartheta +r^{2}}, \qquad 0 < r < 1.

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Problem 7.68

If the function $f(\vartheta )$ is positive in the interval $0 \leq \vartheta < 2\pi$ , then all its associated Toeplitz forms are positive definite. More precisely: If $f(\vartheta )$ lies between the bounds $m \leq f(\vartheta ) \leq M$ in the interval $0 \leq \vartheta < 2\pi$ , then we have

m \leq T_{n}(f) \leq M,

provided that the variables $x_{0}, x_{1}, \ldots , x_{n}$ satisfy the condition $T_{n}(1) = 1$ . The equality sign can hold here only if $f(\vartheta ) = \text{const.}$ at every point of continuity of $f(\vartheta )$ .

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Problem 7.69

Calculate the determinant $D_{n}(f)$ of the form $T_{n}(f)$ for the functions in Problem 67 and show that, if $f(\vartheta )$ is everywhere positive,

\lim _{n \rightarrow \infty } \sqrt[n+1]{D_{n}(f)} = e^{\frac{1}{2\pi } \int _{0}^{2\pi } \log f(\vartheta ) \, d\vartheta } = \mathfrak {G}(f).

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Problem 7.70

Let $f(\vartheta )$ be a trigonometric polynomial of first order and let $h$ be a parameter. The determinant $D_{n}(f-h)$ of the Toeplitz form $T_{n}(f-h)$ associated with $f(\vartheta )-h$ is a polynomial of degree $n+1$ in $h$ with only real zeros. Calculate these zeros.

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Problem 7.71

(continued). The zeros $h_{0n}, h_{1n}, \ldots , h_{nn}$ of $D_{n}(f-h)$ all lie between the minimum and the maximum of $f(\vartheta )$ , that is $m \leq h_{\nu n} \leq M$ , if $m \leq f(\vartheta ) \leq M$ in the interval $0, 2\pi$ . Moreover if $F(h)$ denotes an arbitrary properly integrable function in the interval $m \leq h \leq M$ , then we have

\lim _{n \rightarrow \infty } \frac{F(h_{0n})+F(h_{1n})+\cdots +F(h_{nn})}{n+1} = \frac{1}{2\pi } \int _{0}^{2\pi } F[f(\vartheta )] \, d\vartheta .

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Problem 7.72

Let the Hermitian form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\lambda -\mu } x_{\lambda } \bar{x}_{\mu }, \qquad c_{-\nu } = \bar{c}_{\nu },

be positive definite. Then all the zeros of the polynomial

\begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{-1} & c_{0} & c_{1} & \ldots & c_{n-1} \\ c_{-2} & c_{-1} & c_{0} & \ldots & c_{n-2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{-n+1} & c_{-n+2} & c_{-n+3} & \ldots & c_{1} \\ 1 & z & z^{2} & \ldots & z^{n} \end{vmatrix}

lie in the interior of the unit circle.

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