📂Lebesgue Spaces

Complete Orthonormal Basis and Complete Orthonormal Set

Theorem: Equivalence Conditions of an Orthonormal Set

Let $\left\{ \phi_{n} \right\}_{1}^\infty$ be an orthonormal set of $L^2(a,b)$ and denote $f \in L^2(a,b)$. Then, the following conditions are equivalent.

• $(a)$ For all $n$, if $\left\langle f, \phi_{n} \right\rangle=0$ then $f=0$.

• $(b)$ For all $f\in L^2(a,b)$, the series $\sum_{1}^\infty \left\langle f,\phi_{n}\right\rangle\phi_{n}$ converges to $f$ in the norm sense. That is, the following equation holds:

  $$f=\sum_{1}^\infty \left\langle f,\phi_{n}\right\rangle\phi_{n}$$

• $(c)$ For all $f \in L^2(a,b)$, it satisfies the following equation known as Parseval’s equation:

  $$\| f \|^2 = \sum \limits_{n=1}^{\infty} \left| \left\langle f,\phi_{n} \right\rangle \right|^{2}$$

Explanation

The orthonormal set satisfying $(a) - (c)$ is called an orthonormal basis or a complete orthonormal set.

Observing these three conditions reveals that the orthonormal basis serves a role equivalent to a basis in finite-dimensional vector spaces.

• When $\left\{ \phi_{n} \right\}$ is an orthonormal basis, the constants $\left\langle f, \phi_{n}\right\rangle$ are called (generalized) Fourier coefficients.

• The series $\sum \left\langle f, \phi_{n}\right\rangle\phi_{n}$ is referred to as a (generalized) Fourier series.

Lemma

Assume $f \in L^2(a,b)$ and that $\left\{ \phi_{n} \right\}$ is an orthonormal set in $L^2(a,b)$. Then the series $\sum \left\langle f,\phi_{n} \right\rangle\phi_{n}$ converges in the norm sense. And it satisfies the following inequality:

$$\left\| \sum \left\langle f,\phi_{n}\right\rangle \phi_{n} \right\| \le | f|$$

Proof

• $(a) \implies (b)$

  Assume $(a)$. Then, by the lemma, $\sum \left\langle f, \phi_{n} \right\rangle\phi_{n}$ converges in the norm sense. Let’s define the difference of the series as $g$.

  $$g=f-\sum \limits_{n=1}^{\infty} \left\langle f, \phi_{n} \right\rangle\phi_{n}$$

  Then, it can be shown that $g=0$.

  $$\begin{align*} \left\langle g,\phi_{m} \right\rangle &=\ \left\langle f,\phi_{m}\right\rangle - \sum \limits_{n=1}^{\infty}\left\langle f,\phi_{n} \right\rangle \left\langle \phi_{n}, \phi_{m} \right\rangle \\ &=\ \left\langle f,\phi_{m}\right\rangle - \left\langle f,\phi_{m}\right\rangle \\ &=\ 0 \end{align*}$$

  Therefore, by assumption, $g=0$. Thus, $f= \sum_{n=1}^\infty \left\langle f, \phi_{n} \right\rangle\phi_{n}$

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• $(b) \implies (c)$

  Assume $(b)$. Then, since $f=\sum_{1}^\infty \left\langle f, \phi_{n}\right\rangle\phi_{n}$,

  $$\begin{align*} \| f \|^2 &=\ \left\| \sum \limits_{n=1}^{\infty} \left\langle f, \phi_{n} \right\rangle \phi_{n} \right\| ^2 \\ &= \left\| \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1} ^{N} \left\langle f, \phi_{n} \right\rangle\phi_{n} \right\| ^2 \\ &= \lim \limits_{N \rightarrow \infty} \left\| \sum \limits_{n=1} ^{N} \left\langle f, \phi_{n} \right\rangle\phi_{n} \right\| ^ 2 \\ &= \lim \limits_{N \rightarrow \infty} \sum _{n=1}^{N} | \left\langle f,\phi_{n} \right\rangle |^2 \\ &= \sum \limits _{n=1} ^{\infty} | \left\langle f, \phi_{n} \right\rangle |^2 \end{align*}$$

The third equation holds because the series converges in the norm sense by assumption. The fourth equation is valid due to the Pythagorean theorem.

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• $(c) \implies (a)$

  Assume $(c)$. Then,

  $$\| f \|^2 =\sum \limits _{n=1} ^{\infty}\left| \left\langle f,\phi_{n} \right\rangle \right|^{2}$$

  Therefore, for all $n$, if $\left\langle f, \phi_{n} \right\rangle=0$, then $f=0$ is true.

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