Best uniform (Chebyshev) approximation

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Problem 1457 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.11 "Extrema of a Function. Maximum and Minimum Values of a Function"

Find the "deviation from zero" of the polynomial

P(x)=x(x-1)^{2}(x+2)

on the closed interval $\left[-2,1\right]$ , that is, find

E_{P}=\sup _{-2 \leq x \leq 1}\left|P(x)\right| .

Problem 1458 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.11 "Extrema of a Function. Maximum and Minimum Values of a Function"

What coefficient $q$ should be chosen so that the polynomial

P(x)=x^{2}+q

has the smallest deviation from zero on the closed interval $\left[-1,1\right]$ , that is,

E_{P}=\sup _{-1 \leq x \leq 1}\left|P(x)\right|=\min .

Problem 1459 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.11 "Extrema of a Function. Maximum and Minimum Values of a Function"

The number

\Delta =\sup _{a \leq x \leq b}\left|f(x)-g(x)\right|

is called the absolute difference of the functions $f(x)$ and $g(x)$ on the closed interval $\left[a,b\right]$ .

Find the absolute difference of the functions $f(x)=x^{2}$ and $g(x)=x^{3}$ on the closed interval $\left[0,1\right]$ .

Problem 1460 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.11 "Extrema of a Function. Maximum and Minimum Values of a Function"

On the closed interval $\left[x_{1},x_{2}\right]$ , approximate the function

f(x)=x^{2} ,

by the linear function

g(x)=(x_{1}+x_{2})x+b

so that the absolute difference of $f(x)$ and $g(x)$ (see the problem above) is smallest, and find this smallest absolute difference.

Problem 3710 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

On the interval $(1,3)$ , approximately replace the function $x^{2}$ by a linear function $ax+b$ so that the absolute deviation

\Delta =\sup \left|x^{2}-(ax+b)\right| \quad (1 \leqslant x \leqslant 3)

is minimal.

Problem IX.4.24 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section IX.4 "Fourier Series and Transform"

(a)(b)(c)

Let $f(x)=\sum_{n \geqslant 2} \frac{\cos n x}{n^{2}} \ln n \quad (x \in \mathbb {R})$ . Prove that

(a)

$f \in C^{1}(\mathbb {R}), f \in C^{2}(0,2 \pi ), f^{\prime \prime } \in \mathscr {L}^{1}(0,2 \pi )$ ;

(b)

$\left\| f-S_{n}\right\|_{\infty } \asymp \frac{1}{n \ln n}$ ;

(c)

$E_{n}(f) \asymp \frac{1}{n \ln n}$ . Here $E_{n}(f)=\inf \left\| f-T\right\|_{\infty }$ , where the infimum is taken over all trigonometric polynomials $T$ whose order does not exceed $n$ .

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.