Math Problems Vault

Problem VII.3.1

(a)(b)

Let $n \in \mathbb {N}, r=0,1,2, \ldots$ For $x \in \mathbb {R}$ set

S_{n, r}(x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} x^{k}(1-x)^{n-k}(k-n x)^{r} .

Prove that

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(a)

$S_{n, r+1}(x)=x(1-x)\left(S_{n, r}^{\prime }(x)+n r S_{n, r-1}(x)\right), r \in \mathbb {N}$ ;

(b)

\begin{aligned} & S_{n, 0}(x)=1 ; \quad S_{n, 1}(x)=0 ; \quad S_{n, 2}(x)=n x(1-x) ; \\ & S_{n, 3}(x)=n x(1-x)(1-2 x) ; \\ & S_{n, 4}(x)=n x(1-x)(1+3(n-2) x(1-x)) . \end{aligned}

Problem VII.3.2

(a)(b)(c)

Let $n \in \mathbb {N}, \delta >0$ . For $x \in [0,1]$ set

\sigma _{n, \delta }(x)=\sum _{\substack {0 \leqslant k \leqslant n \\ |k / n-x| \geqslant \delta }} C_{n}^{k} x^{k}(1-x)^{n-k} .

Prove that

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(a)

$\sigma_{n, \delta }(x) \leqslant \frac{1}{4 \delta^{2} n}$ ;

(b)

$\sigma_{n, \delta }(x) \leqslant \frac{1}{4 \delta^{4} n^{2}}$ ;

(c)

$\sigma_{n, \delta }(x) \leqslant \frac{17}{\delta \sqrt{n}} e^{-2 n \delta^{2}}$ .

Problem VII.3.3

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Find the Bernstein polynomials of the following functions:

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(a)

$f_{m}(x)=x^{m}, \quad m=0,1,2$ ;

(b)

$f(x)=a^{x}, a>0$ .

Problem VII.3.4

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let the function $f$ be defined on the interval $\left[0,1\right], n \in \mathbb {N}$ . Prove that

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(a)

$B_{n}(f, 0)=f(0)$ and $B_{n}(f, 1)=f(1)$ ;

(b)

$B_{n}\left(f^{2}, x\right) \geqslant B_{n}^{2}(f, x)$ for all $x \in [0,1]$ .

Problem VII.3.5

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Prove that

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(a)

if the function $f$ is increasing (decreasing) on the interval $\left[0,1\right]$ , then the Bernstein polynomial $B_{n}(f)$ is increasing (decreasing) on $\left[0,1\right]$ ;

(b)

if the function $f$ is convex (concave) on $\left[0,1\right]$ , then $B_{n}(f)$ is a convex (concave) polynomial on $\left[0,1\right]$ .

Problem VII.3.6

(a)(b)(c)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

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

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(a)

$B_{n}(f, x) \leqslant \sup_{\left[0,1\right]} f$ for all $n \in \mathbb {N}$ and $x \in [0,1]$ ;

(b)

if the interval $\Delta$ is contained in the interval $\left[0,1\right]$ , then $\varlimsup_{n \rightarrow +\infty } B_{n}(f, x) \leqslant \sup_{\Delta } f$ for all $x \in \Delta$ ;

(c)

if the intervals $\Delta$ and $\Delta^{\prime }$ are such that $\bar{\Delta } \subset \Delta^{\prime } \subset [0,1]$ and $\sup_{\Delta^{\prime } \backslash \Delta } f<\sup_{\Delta } f$ , then one can specify a sufficiently large number $N=N\left(f, \Delta , \Delta^{\prime }\right)$ such that $B_{n}(f, x)<\sup_{\Delta } f$ for all $x \in \Delta$ and $n>N$ .

Problem VII.3.7

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let the functions $f$ and $g$ be bounded on the interval $\left[0,1\right]$ and coincide in a neighborhood of the point $x_{0} \in [0,1]$ . Prove that the sequences $\left\{ B_{n}\left(f, x_{0}\right)\right\}$ and $\left\{ B_{n}\left(g, x_{0}\right)\right\}$ converge or diverge simultaneously, and in case of convergence have a common limit.

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Problem VII.3.8

(a)(b)(c)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

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

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(a)

if $f$ is continuous at the point $x_{0} \in [0,1]$ , then $B_{n}\left(f, x_{0}\right) \rightarrow f\left(x_{0}\right)$ ;

(b)

if $f$ is continuous at every point of the closed interval $\Delta$ , $\Delta \subset [0,1]$ , then $B_{n}(f) \rightrightarrows f$ on $\Delta$ ;

(c)

if $x_{0} \in (0,1)$ is a point of discontinuity of the first kind of the function $f$ , then

B_{n}\left(f, x_{0}\right) \rightarrow \frac{1}{2}\left(f\left(x_{0}+0\right)+f\left(x_{0}-0\right)\right) .

Problem VII.3.9

(a)(b)

Let $f \in C(\left[0,1\right])$ . Prove that

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(a)

if $\int_{0}^{1} f(x) x^{n} d x=0$ for $n=0,1,2, \ldots$ , then $f \equiv 0$ ;

(b)

if $\int_{0}^{1} f(x)\left(x^{n+1}(1-x)^{2}\right)^{\prime \prime } d x=0$ for all $n \in \mathbb {N}$ , then $f$ is a linear function.

Problem VII.3.10

Let $f \in C(\left[0,1\right])$ and $\int_{0}^{1} f(x) g^{\prime \prime }(x) d x=0$ for every function $g \in C^{2}(\left[0,1\right])$ vanishing in neighborhoods of the points 0 and 1. Prove that $f$ is a linear function.

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Problem VII.3.11

Let $f \in C(\left[0, \pi \right])$ and $\varepsilon >0$ . Prove that there exists a trigonometric polynomial $T$ of the form $T(t)=\sum_{0 \leqslant n \leqslant N} a_{n} \cos n t$ such that $\left|f(t)-T(t)\right|<\varepsilon$ for all $t \in [0, \pi ]$ .

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Problem VII.3.12

(a)(b)

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(a)

Let $f \in C\left(\left[0,1\right]^{2}\right)$ . For $n \in \mathbb {N}$ and $x, y \in \mathbb {R}$ set

\mathscr {B}_{n}(f, x, y)=\sum _{\substack {0 \leqslant k \leqslant n \\ 0 \leqslant j \leqslant n}} C_{n}^{k} C_{n}^{j} f\left(\frac{k}{n}, \frac{j}{n}\right) x^{k} y^{j}(1-x)^{n-k}(1-y)^{n-j}

Prove that $\mathscr {B}_{n}(f) \rightrightarrows f$ on $\left[0,1\right]^{2}$ as $n \rightarrow +\infty$ .

(b)

Let $f \in C\left(\left[0, \pi \right]^{2}\right)$ and $\varepsilon >0$ . Prove that there exists a trigonometric polynomial $T$ of the form

T(u, v)=\sum _{\substack {0 \leqslant k \leqslant N \\ 0 \leqslant j \leqslant N}} a_{k, j} \cos k u \cos j v,

such that $\left|f(u, v)-T(u, v)\right|<\varepsilon$ for all $u, v \in [0, \pi ]$ .

Problem VII.3.13

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let $f \in C^{r}(\left[0,1\right])$ . Prove that $B_{n}^{(r)}(f) \rightrightarrows f^{(r)}$ on $\left[0,1\right]$ .

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Problem VII.3.14

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

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(a)

Let $f \in \operatorname {Lip}_{\alpha }\left[0,1\right]$ . Prove that for all $n \in \mathbb {N}$ and $x \in [0,1]$ the inequality

\left|B_{n}(f, x)-f(x)\right| \leqslant M\left(\frac{x(1-x)}{n}\right)^{\alpha / 2}

holds, where $M$ is some positive constant depending only on the function $f$ .

(b)

Let $\alpha \in (0,1]$ and $f_{\alpha }(x)=\left|x-\frac{1}{2}\right|^{\alpha }$ . Prove that for all $n \in \mathbb {N}$ the inequality

\left|B_{n}\left(f_{\alpha }, \frac{1}{2}\right)-f_{\alpha }\left(\frac{1}{2}\right)\right| \geqslant \frac{1}{4} n^{-\alpha / 2} .

holds.

Problem VII.3.15

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

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(a)

Let the function $f$ be convex on the interval $\left[0,1\right]$ . Prove that $B_{n}(f, x) \downarrow f(x)$ for all $x \in [0,1]$ .

(b)

Let $f \in C(\left[0,1\right])$ and $B_{n}(f, x) \geqslant f(x)$ for all $n \in \mathbb {N}$ and $x \in [0,1]$ . Prove that $f$ is convex on $\left[0,1\right]$ .

Problem VII.3.16

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let $f \in C^{2}(\left[0,1\right])$ . Prove that

B_{n}(f, x)-f(x)=\frac{x(1-x)}{2 n}\left(f^{\prime \prime }(x)+\varepsilon _{n}(x)\right),

where the sequence of functions $\left\{ \varepsilon_{n}\right\}$ converges to zero uniformly on $\left[0,1\right]$ (E. V. Voronovskaya's formula).

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Problem VII.3.17

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let $f \in C(\left[0,1\right]), g \in C^{2}(\left[0,1\right])$ and let the function $g$ be zero outside the interval ( $a, b$ ), where $0<a<b<1$ . Prove that

\sum _{0 \leqslant k \leqslant n}\left(B_{n}\left(f, \frac{k}{n}\right)-f\left(\frac{k}{n}\right)\right) g\left(\frac{k}{n}\right) \rightarrow \frac{1}{2} \int _{0}^{1} f(x)(x(1-x) g(x))^{\prime \prime } d x

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Problem VII.3.18

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Prove that the function $f\left[0,1\right] \rightarrow \mathbb {R}$ is linear if

\sup _{0 \leqslant x \leqslant 1}\left|B_{n}(f, x)-f(x)\right|=o\left(\frac{1}{n}\right) .

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Problem VII.3.19

Let $f \in C(\left[0,1\right])$ . Prove that $f(0)$ and $f(1)$ being integers is equivalent to the function $f$ being the limit of a uniformly convergent (on $\left[0,1\right]$ ) sequence of algebraic polynomials with integer coefficients.

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Problem VII.3.20

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

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

B_{n}(f, x)=f(0)+\sum _{1 \leqslant k \leqslant n} x^{k} C_{n}^{k} \Delta _{k}^{n} f(0),

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 $f$ .

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Problem VII.3.21

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let $f(x)=x^{m}(x \in \mathbb {R}), m=0,1,2, \ldots$ Prove that

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(a)

$B_{n}(f) \rightrightarrows f$ on any finite interval;

(b)

$\left|B_{n}(f, x)\right| \leqslant \max \left(1,\left|x\right|^{m}\right)$ for any $x \in \mathbb {R}$ .

Problem VII.3.22

(a)(b)

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

B_{n}(f, x)=\sum _{0 \leqslant k \leqslant n} C_{n}^{k} f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}

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

Let $f(x)=\sum_{m \geqslant 0} c_{m} x^{m}$ for $x \in (-R, R)$ , where $R>1$ . Prove that

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(a)

$B_{n}(f) \rightrightarrows f$ on any interval $\left[-r, r\right]$ , if $0<r<R$ ;

(b)

if $c_{m} \geqslant 0$ for every $m$ , then $B_{n}(f) \geqslant f$ on $\left[0,1\right]$ and $B_{n}(f) \leqslant f$ on $[1, R)$ .

Problem VII.3.23

(a)(b)(c)

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(a)

Prove that for fixed $n \in \mathbb {N}$ the polynomials $x^{k}(1-x)^{n-k}, k=0,1, \ldots , n$ , form a basis in the space of polynomials of degree not exceeding $n$ .

(b)

Prove that the coefficients $a_{n}^{k}$ of the expansion of a polynomial $P$ of degree $m$ with respect to such bases for $n \geqslant m$ , forming the table

\begin{array}{ccccccc} a_{m}^{0} & a_{m}^{1} & \ldots & a_{m}^{m-1} & a_{m}^{m} & & \\ a_{m+1}^{0} & a_{m+1}^{1} & & & a_{m+1}^{m} & a_{m+1}^{m+1} & \\ \ldots & & & & & & \ldots \\ a_{n}^{0} & a_{n}^{1} & & & & a_{n}^{n-1} & a_{n}^{n} \\ \ldots & & & & & & \ldots \end{array}

satisfy the equalities

a_{n+1}^{0}=a_{n}^{0}, \quad a_{n+1}^{n+1}=a_{n}^{n}, \quad a_{n+1}^{k}=a_{n}^{k-1}+a_{n}^{k} \quad (k=1,2, \ldots , n)

(compare with the equalities for the binomial coefficients $C_{n}^{k}$ ).

(c)

Prove that the condition $P(x)>0$ for $x \in (0,1)$ is necessary and sufficient for the numbers $a_{n}^{k}$ , $k=0,1, \ldots , n$ , to be, for sufficiently large $n$ , not simultaneously zero and nonnegative.