L2 theory of Fourier series

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Problem 11B.7 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11B "Fourier Series and $L^{p}$ of Unit Circle"

Suppose $f \in L^{1}(\partial \mathbf{D})$ . Show that $f \in L^{2}(\partial \mathbf{D})$ if and only if $\sum_{n = -\infty }^{\infty }\left|\widehat{f}(n)\right|^{2} < \infty$ .

Problem 11B.8 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11B "Fourier Series and $L^{p}$ of Unit Circle"

Suppose $f \in L^{2}(\partial \mathbf{D})$ . Prove that $\left|f(z)\right| = 1$ for almost every $z \in \partial \mathbf{D}$ if and only if

\sum _{k = -\infty }^{\infty } \widehat{f}(k) \overline{\widehat{f}(k-n)} = \begin{cases} 1 & \text{ if } n = 0 \\ 0 & \text{ if } n \neq 0\end{cases}

for all $n \in \mathbf{Z}$ .

Problem 4.3.12 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 4.3 "Space of Integrable Functions"

(a)(b)(c)

Let $f$ be a periodic function on $\mathbb {R}$ with period $2\pi$ such that $f|_{[0,2\pi ]}$ belongs to $L^2(0,2\pi )$ . Suppose

f(x) = \sum _{n=-\infty }^{+\infty } a_n e^{inx}.

For each $h \in \mathbb {R}$ define the function $f_h$ by $f_h(x) = f(x-h)$ .

(a)

Give the Fourier expansion of $f_h$ .

(b)

Find the $L^2$ -norm $\left\| f_h - f\right\|_{2}$ in terms of the $a_n$ and $h$ .

(c)

Prove that

\lim _{h \to 0} \frac{\left\| f_h - f\right\| _{2}}{h} = \infty

unless $f$ is constant almost everywhere.

Problem 13.24 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 13 "Fourier Series"

(a)(b)

(a)

Show that if $f \in L^{1}(\mathbf{T})$ and $\widehat{f} \in \ell^{2}(\mathbb {Z})$ , then $f \in L^{2}(\mathbf{T})$ .

(b)

Use part (a) to show that the Plancherel Equality remains true if we assume that $f$ belongs to $L^{1}(\mathbf{T})$ instead of the smaller space $L^{2}(\mathbf{T})$ . In other words, show that if $f \in L^{1}(\mathbf{T})$ , then we have

\sum _{n \in \mathbb {Z}}\left|\widehat{f}(n)\right|^{2} = \left\| f\right\| _{L^{2}}^{2},

in the sense that one side is finite if and only if the other side is finite, and in this case they are equal; otherwise both sides are infinite.

Problem 2.22.80 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.22.2 "Problems"

Let the functions $f$ and $g$ be $2 \pi$ -periodic, with $|f|^{2}$ and $|g|^{2}$ integrable (in the improper sense) on the segment $[-\pi ; \pi ]$ ,

f \sim \sum _{n = -\infty }^{\infty } c_{n} e^{i n x}, \quad g \sim \sum _{n = -\infty }^{\infty } d_{n} e^{i n x}

Prove that then the product $f g$ is a $2 \pi$ -periodic function, absolutely integrable over the period, and if

f g \sim \sum _{n = -\infty }^{\infty } k_{n} e^{i n x}, \quad \text{ then } \quad k_{n} = \sum _{m = -\infty }^{\infty } c_{m} d_{n-m}.

Problem 2.22.139 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.22.2 "Problems"

Let the square of the function $f$ be integrable on the segment $[-\pi ; \pi ]$ . Prove that

\int _{-\pi }^{\pi }\left(f(x)-S_{n+1}(x ; f)\right)^{2} d x \leqslant \int _{-\pi }^{\pi }\left(f(x)-S_{n}(x ; f)\right)^{2} d x, \quad n \in N

Problem 3.20.82 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03b]More problems in Subsection 3.20.2 "Problems"

Prove the stated assertion. Every function $x \in R L_{2}[-\pi ; \pi ]$ (see problem 10) expands into a Fourier series with respect to the trigonometric system of functions, convergent in the mean-square sense (see problem 40):

x(t) = \frac{a_{0}}{2}+\sum _{n = 1}^{\infty }\left(a_{n} \cos n t+b_{n} \sin n t\right)

moreover Parseval's equality

\frac{1}{\pi }\left\| x\right\| ^{2} = \frac{a_{0}^{2}}{2}+\sum _{n = 1}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right)

holds.

Problem 6B.5 from Axler S. "Linear Algebra Done Right" [Axl24]More problems in Section 6B "Orthonormal Bases"

Suppose $f:[-\pi , \pi ] \rightarrow \mathbb {R}$ is continuous. For each nonnegative integer $k$ , define

a_{k} = \frac{1}{\sqrt{\pi }} \int _{-\pi }^{\pi } f(x) \cos (k x) d x \quad \text{ and } \quad b_{k} = \frac{1}{\sqrt{\pi }} \int _{-\pi }^{\pi } f(x) \sin (k x) d x.

Prove that

\frac{a_{0}^{2}}{2}+\sum _{k = 1}^{\infty }\left(a_{k}^{2}+b_{k}^{2}\right) \leq \int _{-\pi }^{\pi } f^{2}

The inequality above is actually an equality for all continuous functions $f:[-\pi , \pi ] \rightarrow \mathbb {R}$ . However, proving that this inequality is an equality involves Fourier series techniques beyond the scope of this book.

Problem 5.4.20 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.4 "Orthogonal Vectors"

With respect to the orthonormal set

B = \left\{ \frac{1}{\sqrt{2\pi }}, \; \frac{\cos t}{\sqrt{\pi }}, \; \frac{\cos 2t}{\sqrt{\pi }}, \ldots , \; \frac{\sin t}{\sqrt{\pi }}, \; \frac{\sin 2t}{\sqrt{\pi }}, \; \frac{\sin 3t}{\sqrt{\pi }}, \ldots \right\} ,

determine the Fourier series expansion of the saw-toothed function defined by $f(t) = t$ for $-\pi < t < \pi$ (extended periodically).

$The periodic extension of the saw-toothed function f(t)=t on (-\pi ,\pi ).$ The periodic extension of the saw-toothed function f(t)=t on (-\pi ,\pi ).

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

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

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

[ККЧ+03b] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[Axl24] Axler S. Linear Algebra Done Right. 4th ed., Springer, 2024.

[Mey00] Meyer C.D. Matrix Analysis and Applied Linear Algebra. SIAM, 2000.