Smooth Functions

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 $n$ times differentiable on $\left[0,1\right]$ and $f^{(k)}(0)= =f^{(k)}(1)=0$ for $k=1, \ldots , n-1$. Prove that for some $x \in [0,1]$ the inequality

holds.

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 $f(x)=P_{1}(x) e^{Q_{1}(x)}+\ldots +P_{n}(x) e^{Q_{n}(x)}$, where $P_{k}, Q_{k} (k=1,2, \ldots , n)$ are real polynomials, $x \in \mathbb {R}$. Prove that if $f \not\equiv 0$, then $f$ has a finite number of zeros.

Let $f \in C^{\infty }([0, \infty ))$, $f(0)=\lim_{x \rightarrow +\infty } f(x)$. Prove that for each $n \in \mathbb {N}$ the function $f^{(n)}$ vanishes.

Let $f \in C^{2}((0,1))$ and $f(x)=o(1)$, $f^{\prime \prime }(x)=O\left(x^{-2}\right)$ as $x \rightarrow +0$. Prove that $f^{\prime }(x)=o\left(x^{-1}\right)$ as $x \rightarrow +0$.

Let $\Delta$ be an interval, $f \in C^{2}(\Delta )$, $M_{k}=\sup_{\Delta }\left|f^{(k)}\right|, k=0,1,2$.

Let $f \in C^{2}([0,+\infty ))$ and $\int_{0}^{+\infty }\left|f(x)\right| d x<+\infty$. Prove the inequalities:

Find $\min \int_{0}^{1}\left|f^{\prime \prime }(x)\right|^{2} d x$ subject to the condition that $f \in C^{2}(\left[0,1\right])$, $f(0)=f(1)=0, f^{\prime }(0)=a$.

Let $f \in C^{2}(\left[0,1\right]), f(0)=f(1)=0$ and $\left|f^{\prime \prime }(x)\right| \leqslant 1$ for all $x \in [0,1]$. Prove the inequalities:

Let $f \in C^{\infty }(\left[a, b\right]), f \not\equiv 0$. Prove that if the function $f$ vanishes on some nonempty interval, then

Does there exist a point $x$ such that $\sup_{n}\left|f^{(n)}(x)\right|=+\infty$?

Let $f \in C^{\infty }(\left[a, b\right])$. Prove that if the sequence of derivatives $\left\{ f^{(n)}(x)\right\}_{n \geqslant 0}$ is bounded at every point $x \in [a, b]$, then it is uniformly bounded: $\left|f^{(n)}(x)\right| \leqslant C$, where $C$ does not depend on $n$ or $x$.

Let $f \in C^{\infty }(\left[a, b\right])$ and suppose that everywhere on $\left[a, b\right]$ the finite limit $\varphi (x)=\lim f^{(n)}(x)$ exists. Prove that the function $\varphi$ has the form $\varphi (x)=$ const $\cdot e^{x}$.

Prove that if at every point of the real line at least one of the derivatives of an infinitely differentiable function vanishes, then this function is a polynomial.

Prove that any function of the form

is infinitely differentiable on $\mathbb {R}$ and that by an appropriate choice of the numbers $c_{n}$ one can arrange for all its derivatives at the point $x=0$ to take any prescribed values in advance (Borel's theorem).

Let $a=\left(a_{1}, \ldots , a_{n}\right) \in \mathbb {R}^{n}, f \in C^{\infty }\left(\mathbb {R}^{n}\right)$. Prove that if $f(a)=0$, then $f$ can be represented in the form

where $g_{j} \in C^{\infty }\left(\mathbb {R}^{n}\right), g_{j}(a)=\frac{\partial f}{\partial x_{j}}(a) \quad (j=1, \ldots , n)$.

Let $a \in \mathbb {R}^{n}$ and let $F$ be a mapping of the set $C^{\infty }\left(\mathbb {R}^{n}\right)$ into $\mathbb {R}$, satisfying three conditions ($f, g \in C^{\infty }\left(\mathbb {R}^{n}\right)$): 1) $F(f+g)=F(f)+F(g)$; 2) $F(\alpha f)=\alpha F(f)(\alpha \in \mathbb {R})$; 3) $F(f g)=F(f) g(a)+f(a) F(g)$.

Prove that $F(f)=\sum_{1 \leqslant j \leqslant n} c_{j} \frac{\partial f}{\partial x_{j}}(a)$ for some $c_{1}, \ldots , c_{n} \in \mathbb {R}$.

Let $f, g$ be nonnegative continuous functions defined on $[0,+\infty )$. Prove that if for some number $C \geqslant 0$ the inequality

holds, then

(Gronwall's inequality). Deduce from this that if the function $h \in C^{1}([0, \infty ))$ and the number $M \geqslant 0$ are such that $h(0)=0$ and $\left|h^{\prime }(t)\right| \leqslant M\left|h(t)\right|$ for $t \geqslant 0$, then $h \equiv 0$.

Let $a \in \mathbb {R}^{n}, A, B \in \mathbb {R}$. Prove that

Prove that if $\mu$ is a finite measure on $\mathbb {R}^{n}$, then the function

(the Fourier transform of the measure $\mu$) is positive definite.

Prove that the functions

are positive definite.

Prove that

Prove that if the function $f: \mathbb {R}^{n} \rightarrow \mathbb {C}$ is positive definite, then

Prove that if the function $f(t)=e^{s \varphi (t)}$ is positive definite for all $s>0$, then the function $\varphi : \mathbb {R}^{n} \rightarrow \mathbb {C}$ is conditionally positive definite.

Let $\Delta =\left[a, b\right], f \in C^{3}(\Delta ), f^{\prime }(x) \neq 0(x \in \Delta )$. The Schwarzian derivative (Schwarzian) $S f$ of the function $f$ is defined by the formula

Verify that for real functions $f, g$ satisfying the conditions listed above, the following statements hold:

Let $R(w, x, y, z)=\frac{(z-w)(y-x)}{(z-y)(x-w)}(w, x, y, z-$ pairwise distinct numbers). Prove that if $f \in C^{1}(\Delta ), f^{\prime }$ is piecewise monotone and for any points $x_{1}<x_{2}<x_{3}<x_{4}$ belonging to the same interval of monotonicity of the function $f$, the inequality

holds, then the function $\left|f^{\prime }\right|$ has no strictly positive local minimum at interior points of the interval $\Delta$.