📂Lebesgue Spaces

Orthogonal Functions and Orthogonal Sets

Definition

Inner Product

The inner product of two functions $f$ and $g$ defined on the interval $[a,b]$ is defined as follows.

$$\braket{f , g} := \int_{a}^b f(x) g(x) dx$$

When $f, g$ is a complex function, then,

$$\braket{f, g} := \int_{a}^{b} f(x) \overline{g(x)} dx$$

In this case, $\overline{z}$ is the conjugate complex of $z$.

Orthogonal Functions

Two complex functions $f$, $g$ are said to be orthogonal on the interval $[a,b]$ if they satisfy the following equation.

$$\braket{f, g} = \int_{a}^{b} f(x) \overline{g(x)} dx = 0$$

Since we defined the inner product of two functions as an integral, it is natural to say that they are orthogonal when the integral equals $0$.

Orthogonal Set

Functions $\phi_{1}$, $\phi_2$, $\phi_{3}$, $\dots$ are said to form an orthogonal set if they satisfy the following equation.

$$\braket{\phi_{m}, \phi_{n}} = \int_{a}^b \phi_{m} (x) \overline{ \phi_{n}(x) } dx = 0 \quad (m\ne n)$$

Normalization

The norm of function $f$ is defined as follows.

$$\left\| f \right\| := \sqrt{\braket{f, f}} = \left( \int_{a}^b \left| f(x) \right| ^2 dx \right) ^{ \frac{1}{2} }$$

To normalize a function $f$ means to multiply it by an appropriate constant so that the norm of $f$ becomes $1$. The normalized function $f_{\text{normal}}$ of $f$ is,

$$f_{\mathrm{normal}} = \frac{1}{ \left\| f \right\| }f$$

Orthonormal Set

An orthogonal set $\left\{ \phi_{1}, \phi_{2}, \cdots \right\}$ whose elements are all normalized functions is called an orthonormal set. That is, for all $n, m$, it satisfies the following.

$$\braket{\phi_{m}, \phi_{n}} = \int_{a}^b \phi_{m} (x) \overline{ \phi_{n}(x) } dx=\delta_{mn}$$

Here, $\delta_{mn}$ is the Kronecker delta.