Explicit orthonormal bases of sequence and Sobolev-type spaces

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Problem 8C.10 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 8C "Orthonormal Bases"

(a)(b)(c)(d)

(a)

Show that the orthonormal family given in the first bullet point of Example 8.51 is an orthonormal basis of $\ell^{2}$ .

(b)

Show that the orthonormal family given in the second bullet point of Example 8.51 is an orthonormal basis of $\ell^{2}(\Gamma )$ .

(c)

Show that the orthonormal family given in the fourth bullet point of Example 8.51 is not an orthonormal basis of $L^{2}([0,1))$ .

(d)

Show that the orthonormal family given in the fifth bullet point of Example 8.51 is not an orthonormal basis of $L^{2}(\mathbb {R})$ .

Remark.

Example 8.51:

(a)

First bullet point: for $k \in \mathbb {N}$ , let $e_{k} \in \ell^{2}$ be the sequence all of whose coordinates are $0$ except for the $k$ th coordinate, which is $1$ . Then $\left\{ e_{k}\right\}_{k \in \mathbb {N}}$ is an orthonormal family in $\ell^{2}$ .

(b)

Second bullet point: for a nonempty set $\Gamma$ , with $\ell^{2}(\Gamma ) := L^{2}(\mu )$ for $\mu$ counting measure on $\Gamma$ , define $e_{k}: \Gamma \rightarrow \mathbf{F}$ for $k \in \Gamma$ by $e_{k}(j) = 1$ if $j = k$ and $e_{k}(j) = 0$ if $j \neq k$ . Then $\left\{ e_{k}\right\}_{k \in \Gamma }$ is an orthonormal family in $\ell^{2}(\Gamma )$ .

(c)

Fourth bullet point: for $k$ a nonnegative integer, define $e_{k}: [0,1) \rightarrow \mathbf{F}$ (a Rademacher function) by $e_{k}(x) = 1$ if $x \in \left[\frac{n-1}{2^{k}}, \frac{n}{2^{k}}\right]$ for some odd integer $n$ , and $e_{k}(x) = -1$ if $x$ is in such an interval for some even integer $n$ . Then $\left\{ e_{k}\right\}_{k \in \left\{ 0,1,\ldots \right\} }$ is an orthonormal family in $L^{2}([0,1))$ .

(d)

Fifth bullet point: for $k$ a nonnegative integer and $m \in \mathbb {Z}$ , define $e_{k,m}: \mathbb {R} \rightarrow \mathbf{F}$ by translating the function from the fourth bullet point by $m$ : $e_{k,m}(x) = 1$ if $x \in m + \left[\frac{n-1}{2^{k}}, \frac{n}{2^{k}}\right]$ for some odd integer $n \in [1, 2^{k}]$ , $e_{k,m}(x) = -1$ if $x$ is in such a translated interval for some even integer $n \in [1, 2^{k}]$ , and $e_{k,m}(x) = 0$ if $x \notin [m, m+1)$ . Then $\left\{ e_{k,m}\right\}_{(k,m) \in \left\{ 0,1,\ldots \right\} \times \mathbb {Z}}$ is an orthonormal family in $L^{2}(\mathbb {R})$ .

Problem 2.6 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 2.2 "Problems to Solve"

(1)(2)(3)

Let $\alpha = \left(\alpha_{1}, \alpha_{2}, \ldots \right)$ , where $\alpha_{k} > 0, k \geq 1$ . Consider the set $l_{2, \alpha }$ of all numerical sequences $x = \left(x_{1}, x_{2}, \ldots \right)$ for which $\sum_{k = 1}^{\infty } \alpha_{k}\left|x_{k}\right|^{2} < \infty$ . Verify that $l_{2, \alpha }$ is a Hilbert space with the scalar product $(x, y) = \sum_{k = 1}^{\infty } \alpha_{k} x_{k} \overline{y_{k}}, x, y \in l_{2, \alpha }$ .

Construct an orthonormal basis in the space $l_{2, \alpha }$ in the following cases:

(1)

$\alpha_{k} = k, k \geq 1$ ;

(2)

$\alpha_{k} = k^{2}, k \geq 1$ ;

(3)

$\alpha_{k} = e^{-k}, k \geq 1$ .

Problem 3.57 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 3.3 "Orthogonal and orthonormal systems and bases in Hilbert spaces"

Is the orthonormal system obtained in the previous problem a basis in the space $H: = \left\{ f \in W_{2}^{1}[-\pi , \pi ]: f(-\pi ) = f(\pi )\right\}$ ?

Problem 3.59 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 3.3 "Orthogonal and orthonormal systems and bases in Hilbert spaces"

Prove that the set of sequences of Fourier coefficients of all functions from the space $\dot{W}_{2}^{1}[0,1]$ with respect to the system $\left\{ \varphi_{n}\right\}$ from the previous problem coincides with the linear space $l_{2, \mathbb {N}}$ (see the definition in problem 3.13).

Problem I.4.2 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section I.4 "Orthonormal Sets of Vectors and Bases"

Verify the statements in Example 4.4: if $\mathcal{H} = \mathbb {F}^d$ and for $1 \leq k \leq d$ , $e_k =$ the $d$ -tuple with $1$ in the $k$ th place and zeros elsewhere, then $\left\{ e_1, \ldots , e_d\right\}$ is a basis for $\mathcal{H}$ .

Problem I.4.3 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section I.4 "Orthonormal Sets of Vectors and Bases"

Verify the statements in Example 4.5: let $\mathcal{H} = l^2(I)$ as in Example 1.7. For each $i$ in $I$ define $e_i$ in $\mathcal{H}$ by $e_i(i) = 1$ and $e_i(j) = 0$ for $j \neq i$ . Then $\left\{ e_i : i \in I\right\}$ is a basis.

Problem I.4.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section I.4 "Orthonormal Sets of Vectors and Bases"

Find an infinite orthonormal set in the Hilbert space of Example 1.8 (the collection of all absolutely continuous functions $f : [0,1] \to \mathbb {F}$ with $f(0) = 0$ and $f' \in L^2(0,1)$ , with $\left\langle f, g \right\rangle = \int_0^1 f'(t)\overline{g'(t)} \, dt$ ).

Problem 3.20.76 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.20.2 "Problems"

Prove the stated assertion. The elements $e_{n} = \left(x_{1}^{(n)} ; \ldots ; x_{m}^{(n)} ; \ldots \right), x_{n}^{(n)} = 1, x_{m}^{(n)} = 0$ for $m \neq n, n = 1,2, \ldots$ , form an orthonormal basis in the space $l_{2}$ (see problem 15).

Problem 6B.2 from Axler S. "Linear Algebra Done Right" [Axl24]More problems in Section 6B "Orthonormal Bases"

(a)(b)

(a)

Suppose $\theta \in \mathbb {R}$ . Show that both

(\cos \theta , \sin \theta ),(-\sin \theta , \cos \theta ) \quad \text{ and } \quad (\cos \theta , \sin \theta ),(\sin \theta ,-\cos \theta )

are orthonormal bases of $\mathbb {R}^{2}$ .

(b)

Show that each orthonormal basis of $\mathbb {R}^{2}$ is of the form given by one of the two possibilities in (a).

References

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

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[ККЧ+03] 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.