Recall the space \(PC(a,b)\) of piecewise continuous functions on the interval \((a,b)\). There is a strong analogy between \(PC(a,b)\) and \(\mathbb{C}^k\), if we think of functions \(f \in PC(a,b)\) as infinite-dimensional vectors.
The inner product of two functions \(f, g \in PC(a,b)\) is defined by \[ \langle f, g \rangle = \int_a^b f(x) \overline{g(x)} \, dx, \] where \(\overline{g(x)}\) denotes the complex conjugate of \(g(x)\).
The norm of a function is defined by \[ \| f \| = \sqrt{\langle f, f \rangle} = \sqrt{\int_a^b |f(x)|^2 \, dx}. \]
Properties of inner products and norms
The inner product has the following properties:
Linearity in the first argument: For all \(f, g, h \in PC(a,b)\) and \(c, d \in \mathbb{C}\), we have \[ \langle cf + dg, h \rangle = c \langle f, h \rangle + d \langle g, h \rangle. \]
Conjugate linearity in the second argument: For all \(f, g, h \in PC(a,b)\) and \(c, d \in \mathbb{C}\), we have \[ \langle f, cg + dh \rangle = \overline{c} \langle f, g \rangle + \overline{d} \langle f, h \rangle. \]
Conjugate symmetry: For all \(f, g \in PC(a,b)\), we have \[ \langle f, g \rangle = \overline{\langle g, f \rangle}. \]
The norm has the following properties:
Positive definiteness: For all \(f \in PC(a,b)\), we have \(\| f \| \geq 0\), and \(\| f \| = 0\) if and only if \(f = 0\).
Homogeneity: For all \(f \in PC(a,b)\) and \(c \in \mathbb{C}\), we have \(\| cf \| = |c| \| f \|\).
Lemma 3.1 (Polarization Identity)
For any \(f, g \in PC(a,b)\), we have \[ \| f + g \|^2 = \| f \|^2 + 2\operatorname{Re}{\langle f, g \rangle} + \| g \|^2, \] where \(\operatorname{Re}{z}\) denotes the real part of a complex number \(z\).
Cauchy-Schwarz Inequality
For any \(f, g \in PC(a,b)\), we have \[ |\langle f, g \rangle| \leq \| f \| \| g \|. \]
Triangle Inequality
For any \(f, g \in PC(a,b)\), we have \[ \| f + g \| \leq \| f \| + \| g \|. \]
Pythagorean Theorem
If \(f_1,f_2, \ldots, f_n\) are mutually orthogonal functions in \(PC(a,b)\), then \[ \| f_1 + f_2 + \cdots + f_n \|^2 = \| f_1 \|^2 + \| f_2 \|^2 + \cdots + \| f_n \|^2. \]
A function \(f\) is normalized, or is a unit vector if \(\| f\|=1\).
A collection of nonzero functions \(\{f_1, f_2, \ldots\}\) is an orthogonal set if \(\langle f_j, f_k \rangle = 0\) for all \(j \neq k\). It is an orthonormal set if it is an orthogonal set and each \(f_j\) is a unit vector.
Recall Theorem 3.1 for vectors, from the previous section:
Theorem 3.1
Let \(\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_k\}\) be an orthonormal set of vectors in \(\mathbb{C}^k\). Then for any \(\mathbf{a} \in \mathbb{C}^k\), we have
\[ \mathbf{a} = \langle \mathbf{a}, \mathbf{u}_1 \rangle \mathbf{u}_1 + \langle \mathbf{a}, \mathbf{u}_2 \rangle \mathbf{u}_2 + \cdots + \langle \mathbf{a}, \mathbf{u}_k \rangle \mathbf{u}_k, \]
and \[ \| \mathbf{a} \|^2 = |\langle \mathbf{a}, \mathbf{u}_1 \rangle|^2 + |\langle \mathbf{a}, \mathbf{u}_2 \rangle|^2 + \cdots + |\langle \mathbf{a}, \mathbf{u}_k \rangle|^2. \]
Is there an analogue of Theorem 3.1 for functions in \(PC(a,b)\)?
We must be careful here, because \(PC(a,b)\) is an infinite-dimensional vector space.
Consider the functions
\[ \phi_n(x) = \frac{1}{\sqrt{2\pi}} e^{inx}, \quad n \in \mathbb{Z}. \]
Discuss an analog of Theorem 3.1 for the set \(\{\phi_n\}_{n \in \mathbb{Z}}\) in \(PC(-\pi,\pi)\).
Consider the functions
\[ \psi_0(x) = \frac{1}{\sqrt{\pi}}, \quad \psi_n(x) = \sqrt{\frac{2}{\pi}} \cos{nx}, \quad n \in \mathbb{N}. \]
Discuss an analog of Theorem 3.1 for the set \(\{\psi_n\}_{n=0}^{\infty}\) in \(PC(0,\pi)\).