Increments and finite differences of functions

For $f$ a real valued function on the real line, define the function $\Delta f$ by $\Delta f(x) = f(x+1)-f(x)$. For $n \geq 2$, define $\Delta^{n} f$ recursively by $\Delta^{n} f = \Delta (\Delta^{n-1} f)$. Prove that $\Delta^{n} f = 0$ if and only if $f$ has the form $f(x) = a_{0}(x)+a_{1}(x)x+\cdots +a_{n-1}(x)x^{n-1}$ where $a_{0}, a_{1}, \ldots , a_{n-1}$ are periodic functions of period 1.

Let

and

be, respectively, the first- and second-order finite differences of the function $f(x)$.

Prove that if the function $f(x)$ ($-\infty <x<+\infty$) is continuous and

then this function is linear, i.e.,

where $a$ and $b$ are constants.

If $x$ changes from $1$ to $1000$, find the increment $\Delta x$ of the independent variable $x$ and the corresponding increment $\Delta y$ of the function $y=\lg x$.

If $x$ changes from $0.01$ to $0.001$, find the increment $\Delta x$ of the independent variable $x$ and the corresponding increment $\Delta y$ of the function $y=\frac{1}{x^{2}}$.

Let:

Prove that:

For the function $y = x^{3} - 3x^{2} + 5x - 7$ at $x=0$, the values of the differences $\Delta y_{0} = +3$, $\Delta^{2} y_{0} = 0$, $\Delta^{3} y_{0} = 5$ have been computed, corresponding to an increment $h_{0}=1$ of the argument. It is required to compute the differences $\overline{\Delta y_{0}}$, $\overline{\Delta^{2} y_{0}}$, $\overline{\Delta^{3} y_{0}}$, corresponding to an increment of the argument $h_{1} = \dfrac {1}{10}$.

For the function $y = 2x^{4} - x^{3} + 3x^{2} - 1$ at $x=0$, the differences $\Delta y_{0} = 4$, $\Delta^{2} y_{0} = 28$, $\Delta^{3} y_{0} = 66$, $\Delta^{4} y_{0} = 48$ have been computed, corresponding to an increment $h_{0}=1$ of the argument. It is required to compute the differences $\overline{\Delta y_{0}}$, $\overline{\Delta^{2} y_{0}}$, $\overline{\Delta^{3} y_{0}}$, $\overline{\Delta^{4} y_{0}}$, corresponding to an increment of the argument $h_{1} = \dfrac {1}{10}$.

Prove that if the function $f$ is $m$ times differentiable on the interval $(a, b)$ and $x, x+m h \in (a, b)$, then

where $0<\theta <m$ (for the definition of $\Delta_{h}^{m} f(x)$ see problem III.3.1).

Let the function $f$ be continuous on $\mathbb {R}$, and suppose there exist real numbers $a_{k}, b_{k}, c_{k}(1 \leqslant k \leqslant n)$ such that

Prove that if $c_{k} \neq 0$ for all $k$, then $f$ is a polynomial.

Let $n \in \mathbb {N}, f:\left[0,1\right] \rightarrow \mathbb {R}$. The polynomial

is called the Bernstein polynomial of the function $f$.

Let the function $f$ be defined on the interval $\left[0,1\right]$. Prove that

where $\quad \Delta_{1}^{n} f(y)=f\left(y+\frac{1}{n}\right)-f(y), \quad \Delta_{k+1}^{n} f(y)=\Delta_{1}^{n}\left(\Delta_{k}^{n} f(y)\right), \quad k \in \mathbb {N}$. In particular, if $f$ is a polynomial, then the degree of $B_{n}(f)$ does not exceed the degree of $f$.