Exercise 6, § 3

Functions of a Real Variable, I, 40

Let $\mathbf{f}$ be a vector function defined on an open interval $I \subset \mathbf{R}$ , taking values in a normed space E.

Put $\Delta \mathbf{f}(x; h_1) = \mathbf{f}(x + h_1) - \mathbf{f}(x)$ , and then, inductively, define

\Delta^p \mathbf{f}(x; h_1, h_2, \ldots, h_{p-1}, h_p) = \Delta^{p-1} \mathbf{f}(x + h_p; h_1, \ldots, h_{p-1}) - \Delta^{p-1} \mathbf{f}(x; h_1, \ldots, h_{p-1});

these functions are defined for each $x \in I$ when the $h_i$ are small enough.

a) If the function $f$ is $n$ times differentiable at the point $x$ (and so $n - 1$ times differentiable on a neighbourhood of $x$ ), one has

\lim_{(h_1, \ldots, h_n) \to (0, \ldots, 0)} \frac{\Delta^n f(x; h_1, \ldots, h_n)}{h_1 h_2 \ldots h_n} = f^{(n)}(x)

(argue by induction on

n

, using the mean value theorem).

b) If $f$ is $n$ times differentiable on the interval $I$ , one has

\| \Delta^n f(x; h_1, \ldots, h_n) - f^{(n)}(x_0) h_1 h_2 \ldots h_n \| \\ \leq |h_1 h_2 \ldots h_n| \sup \| f^{(n)}(x + t_1 h_1 + \cdots + t_n h_n) - f^{(n)}(x_0) \|

the supremum being taken over the set of

(t_i)

such that

0 \leq t_i \leq 1

for

1 \leq i \leq n

(same method).

c) If $f$ is a real function which is $n$ times differentiable on $I$ , one has

\Delta^n f(x; h_1, h_2, \ldots, h_n) = h_1 h_2 \ldots h_n f^{(n)}(x + \theta_1 h_1 + \cdots + \theta_n h_n)

the numbers

\theta_i

belonging to

[0, 1]

(same method, using I, p. 22, corollary).

Solution

No solution has been written yet.