Dirichlet and Fejer kernels

$a_{\nu }$ , $b_{\nu }$ being the Fourier coefficients of the function $f(x)$ . Prove that $\left|\sigma_{n}(x)\right| < M$ , if $\left|f(x)\right| < M$ throughout the period.

Problem 4911 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.9 "Fourier Series and Related Questions"

Prove that if $f(x)$ is a continuous function on the interval $\left[-\pi ,\pi \right]$ , $l \leqslant f(x) \leqslant L$ , and for every $m$ , $\left|ma_{m}\right| < A$ , $\left|mb_{m}\right| < B$ , then

l-(A+B) \leqslant s_{n}(x) \leqslant L+(A+B);

where $a_{m}$ , $b_{m}$ are the Fourier coefficients, $s_{n}(x)$ is the $n$ -th partial sum of the Fourier series of $f(x)$ .

Problem 13.19 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove Lemma 13.17.

Remark.

Lemma 13.17: Let $\chi_{N}$ denote the discrete characteristic function of the set $\left\{ -N, \ldots , N\right\}$ , i.e., $\chi_{N}(n) = \sum_{m = -N}^{N}\delta_{m}(n)$ (so $\chi_{N}(n) = 1$ for $-N \leq n \leq N$ and $\chi_{N}(n) = 0$ for $\left|n\right| > N$ ). Then for each $N \in \mathbb {N}$ we have $d_{N}(x) = \sum_{n = -N}^{N} e^{2 \pi i n x} = \frac{\sin \left((2 N+1)\pi x\right)}{\sin \pi x} = \widehat{\chi }_{N}(x) = \left(\chi_{N}\right)^{\vee }(x)$ .

Problem 13.20 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove Theorem 13.18.

Remark.

Theorem 13.18: $d_{N} \in L^{1}(\mathbf{T})$ and $\int_{0}^{1} d_{N} = 1$ for each $N \in \mathbb {N}$ , and for $N > 1$ we have $\frac{4}{\pi^{2}} \sum_{k = 1}^{N} \frac{1}{k} \leq \left\| d_{N}\right\|_{L^{1}} \leq 3+\frac{4}{\pi^{2}} \sum_{k = 1}^{N} \frac{1}{k}$ .

Problem 13.22 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

Prove the equality appearing in equation (13.18).

Problem 13.23 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

(a)(b)

Prove Lemma 13.20.

(a)

$w_{N}(x)$ has the closed form $w_{N}(x) = \frac{1}{N+1}\left(\frac{\sin \pi (N+1) x}{\sin \pi x}\right)^{2}$ .

(b)

$\left\{ w_{N}\right\}$ forms an approximate identity on $\mathbf{T}$ .

Problem 502 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.5.2 "Operators in Hilbert Space"

Let $A$ be a nuclear integral operator on $L^{2}\left({[0,1]}\right)$ with continuous kernel $K(x,y)$ . Prove the identity

\operatorname {tr}A=\int _0^1K(x,x)\, dx.

Problem 519 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

Construct a $\delta$ -shaped sequence consisting of trigonometric polynomials on the torus $\mathrm{T}^n$ .

Problem 2.6.116 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.6.2 "Problems"

Prove that $\int_{0}^{\pi / 2}\left(\frac{\sin n x}{\sin x}\right)^{2} d x = \frac{n \pi }{2}, \; n \in N$ (Fejér).

Problem 2.6.205 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.6.2 "Problems"

(1)(2)

Find the integrals:

(1)

$\int_{0}^{\pi } \frac{\sin n x}{\sin x} d x$ ;

(2)

$\int_{0}^{\pi } \frac{\cos (2 n+1) x}{\cos x} d x$ .

Problem 2.6.219 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.6.2 "Problems"

Prove the equality

\int _{0}^{\pi / 2} \frac{\sin n x}{\sin x} d x = \frac{\pi }{2}-2 \cos \frac{\pi n}{2} \int _{0}^{1} \frac{x^{n} d x}{1+x^{2}}, \quad n = 0,1,2, \ldots

Problem 2.134 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Singular Integrals. Weierstrass' Approximation Theorem"

Let $f(x)$ be a continuous periodic function with period $2\pi$ . Then

\lim _{n \to \infty } \frac{1}{2n\pi } \int _{0}^{2\pi } f(t) \left(\frac{\sin n \dfrac {x-t}{2}}{\sin \dfrac {x-t}{2}}\right)^{2} \, dt = f(x),

the convergence is uniform for all $x$ .

References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.