Orthogonality of the trigonometric system

Verify that the family $\left\{ e_{k}\right\}_{k \in \mathbf{Z}}$ as defined in the third bullet point of Example 8.51 is an orthonormal family in $L^{2}((-\pi , \pi ])$. The following formulas should help:

Using Euler's formula

prove that:

($n$ and $m$ are integers).

Prove that the system of functions $e_n(x)=e^{2\pi inx}$, $n\in \mathbb {Z}$, is a Hilbert basis in $L^{2}\left({(0,1)}\right)$.

Prove that

(If a function $f$ has the form $f(x) = u(x)+i v(x)$, where the functions $u(x)$ and $v(x)$ take real values and are integrable on the interval $[a ; b]$, then by definition $\int_{a}^{b} f(x) d x = \int_{a}^{b} u(x) d x+i \int_{a}^{b} v(x) d x$.)

The trigonometric system of functions

is orthogonal in the space $L_{2}[-\pi ; \pi ]$ (see problem 35), and the corresponding orthonormal system has the form

The sequence of functions $\sin (2 n-1) t / 2, n = 1,2, \ldots$, forms an orthogonal system in the space $L_{2}[0 ; \pi ]$ (see problem 35).

Prove the stated assertion. The functions $1, \sin \frac{2 \pi n(t-a)}{b-a}, \cos \frac{2 \pi n(t-a)}{b-a}, \quad n = 1,2, \ldots$, are orthogonal in the space $C L_{2}^{1}[a ; b]$ (see problem 36).

With $z=e^{i\vartheta }$ we find for $k,l=0,1,2,3,\ldots$

The functions $1,z,z^{2},z^{3},\ldots$ form an orthogonal system on the unit circle.

Show that the trigonometric system

is an orthonormal sequence in $L^{2}([-\pi , \pi ], m)$.

Suppose $n$ is a positive integer. Prove that

is an orthonormal list of vectors in $C[-\pi , \pi ]$, the vector space of continuous real-valued functions on $[-\pi , \pi ]$ with inner product

The classical orthonormal basis for $L^{2}[0,2 \pi ]$ with Lebesgue measure is the trigonometric system (19.17) $(2 \pi )^{-1}, \quad \pi^{-1 / 2} \sin n x, \quad \pi^{-1 / 2} \cos n x, \quad n=1,2, \ldots$

Prove orthonormality. Hint: Express the sines and cosines in terms of $e^{i n x} \pm e^{-i n x}$, multiply out the products, and use the fact that $\int_{0}^{2 \pi } e^{i m x} d x$ is $2 \pi$ or 0 as $m=0$ or $m \neq 0$. (For the completeness of the trigonometric system, see Problem 26.26.)