Question

Let \(C[-\pi, \pi]^{n}\) denote the space of complex-valued continuous functions defined on the cube \([-\pi, \pi]^{n}\). On this linear space we define the inner product $$ \langle f, g\rangle=\frac{1}{(2 \pi)^{n}} \underbrace{\int_{-\pi}^{\pi} \ldots \int_{-\pi}^{\pi}}_{n \text { times }} f\left(x_{1}, \ldots, x_{n}\right) \overline{g\left(x_{1}, \ldots, x_{n}\right)} d x_{1} \cdots d x_{n} . $$ Prove that the family of functions $$ \left\\{e^{i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)}\right\\}_{m_{1}, \ldots, m_{n} \in \mathbb{Z}} $$ is an orthonormal system with respect to the above inner product. If we define $$ c_{m_{1}, \ldots, m_{n}}=\frac{1}{(2 \pi)^{n}} \underbrace{\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} f\left(x_{1}, \ldots, x_{n}\right) e^{-i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)} d x_{1} \cdots d x_{n}} $$ then the series $$ \sum_{m_{1}, \ldots, m_{n}=-\infty}^{\infty} c_{m_{1}, \ldots, m_{n}} e^{i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)} $$ is called the multivariate complex Fourier series of \(f\). Prove that if \(f_{1}, f_{2}\), \(\ldots, f_{n}\) are functions in \(C[-\pi, \pi]\) and $$ f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f_{1}\left(x_{1}\right) f_{2}\left(x_{2}\right) \cdots f_{n}\left(x_{n}\right) $$ then $$ c_{m_{1}, \ldots, m_{n}}=a_{m_{1}}^{1} a_{m_{2}}^{2} \cdots a_{m_{n}}^{n} $$ where the \(a_{m}^{k}\) are the coefficients of the complex Fourier series of \(f_{k}\). That is, \(f_{k}(x) \sim \sum_{m=-\infty}^{\infty} a_{m}^{k} e^{i m x}\).

Short answer

The exponential functions form an orthonormal system, and the coefficients factor as \( c_{m_1,\ldots,m_n} = a_{m_1}^1 a_{m_2}^2 \cdots a_{m_n}^n \).

Step-by-step solution

Define Orthonormality

To prove the orthonormality of the given family of functions, we need to show two key properties: orthogonality and norm. For orthogonality, we must show \( \langle e^{i(m_1 x_1 + \cdots + m_n x_n)}, e^{i(n_1 x_1 + \cdots + n_n x_n)} \rangle = 0 \) when \((m_1, \ldots, m_n) eq (n_1, \ldots, n_n)\). For norm, we show \( \langle e^{i(m_1 x_1 + \cdots + m_n x_n)}, e^{i(m_1 x_1 + \cdots + m_n x_n)} \rangle = 1 \).

Verify Orthogonality

For different frequency indices \( (m_1, \ldots, m_n) \) and \( (n_1, \ldots, n_n) \), compute the inner product: \[ \langle e^{i(m_1 x_1 + \cdots + m_n x_n)}, e^{i(n_1 x_1 + \cdots + n_n x_n)} \rangle = \frac{1}{(2\pi)^n} \int_{[-\pi, \pi]^n} e^{i((m_1-n_1)x_1 + \cdots + (m_n-n_n)x_n)} \, dx_1 \cdots dx_n. \] Since the integrand is a product of complex exponentials with different frequencies, the integral evaluates to zero, proving orthogonality.

Check Norm

To verify the norm, compute: \[ \langle e^{i(m_1 x_1 + \cdots + m_n x_n)}, e^{i(m_1 x_1 + \cdots + m_n x_n)} \rangle = \frac{1}{(2\pi)^n} \int_{[-\pi, \pi]^n} e^{0} \, dx_1 \cdots dx_n = 1. \] Since the exponent is zero, the integral evaluates to one, confirming orthonormality. Together with Step 2, this shows the system is orthonormal.

Express Function as Product

Given a function \( f(x_1, x_2, \ldots, x_n) = f_1(x_1) f_2(x_2) \cdots f_n(x_n) \), we want to express its Fourier series coefficients \( c_{m_1, \ldots, m_n} \) in terms of the individual Fourier coefficients \( a_m^k \) of \( f_k \).

Compute the Coefficients

Compute \( c_{m_1, \ldots, m_n} \): \[ c_{m_1, \ldots, m_n} = \frac{1}{(2\pi)^n} \int_{[-\pi, \pi]^n} f_1(x_1) f_2(x_2) \cdots f_n(x_n) e^{-i(m_1 x_1 + \cdots + m_n x_n)} \, dx_1 \cdots dx_n. \] This integral can be factored as a product of integrals, each resembling the coefficient in a single-variable Fourier series.

Factorize the Integral

Factor the integral: \[ c_{m_1, \ldots, m_n} = \left( \frac{1}{2\pi} \int_{-\pi}^{\pi} f_1(x_1)e^{-im_1 x_1} \, dx_1 \right) \cdots \left( \frac{1}{2\pi} \int_{-\pi}^{\pi} f_n(x_n)e^{-im_n x_n} \, dx_n \right). \] Thus, \( c_{m_1, \ldots, m_n} = a_{m_1}^1 a_{m_2}^2 \cdots a_{m_n}^n \), where \( a_m^k \) are the Fourier coefficients of \( f_k \).

Final Step: Conclude

By having established both the orthonormality of the exponential functions and the product formula for the coefficients, we've proven the desired properties of the Fourier series representation.

Key concepts

Orthogonormality

The term orthonormality combines the concepts of orthogonality and normalization within a set of functions. A set of functions is orthonormal if the functions are pairwise orthogonal and each function has a unit norm.

• **Orthogonality**: Two functions are orthogonal if their inner product is zero. In the context of the multivariate Fourier series, for the two functions \( e^{i(m_1 x_1 + \cdots + m_n x_n)} \) and \( e^{i(n_1 x_1 + \cdots + n_n x_n)} \), orthogonality means their inner product equals zero when the tuples \( (m_1, \ldots, m_n) \) and \( (n_1, \ldots, n_n) \) are different.
• **Normalization**: A function is said to be normalized if its inner product with itself equals one. For our exponential functions, this means the integral of one of these functions times its complex conjugate over the defined space is one.

These properties ensure that each member of the function family is independent and unique, providing a basis that simplifies many computations in function space.

Complex-Valued Continuous Functions

Complex-valued continuous functions are functions where both input and output can take complex numbers. In this context, we consider functions defined on the interval \([-\pi, \pi]\) which form a crucial part of solving multivariate Fourier series problems.

• **Complex Numbers**: These include both real and imaginary parts, typically written as \( a + bi \), where \( i^2 = -1 \).
• **Continuity**: A function is continuous if small changes in the input result in small changes in the output, meaning that there are no sudden jumps or breaks.
• **Domain and Range**: For multivariate Fourier series, the domain is typically a cube \([-\pi, \pi]^n\), and the range is the set of all complex numbers.

These functions are fundamental for constructing and analyzing Fourier series, particularly because their richness allows almost any periodic signal to be decomposed into a series of orthonormal functions.

Inner Product Space

An inner product space extends the concept of dot product to function space. It's a mathematical setting that combines geometry with the algebra of functions. The inner product is denoted as \( \langle f, g \rangle \), providing a measure of similarity or relationship between two functions.

• **Linear Space**: Refers to a mathematical space where functions can be added together and multiplied by scalars, similar to vector addition and scalar multiplication in linear algebra.
• **Complex Conjugate**: When dealing with complex-valued functions, we use the complex conjugate \( \overline{g(x)} \) during integration to ensure real values for the inner product. This involves changing the sign of the imaginary part of a complex number.
• **Orthogonality and Length**: The inner product defines orthogonality (zero inner product) and norm (square root of inner product with itself) as key properties.

The inner product space concept is integral in finding the orthogonal projections of functions onto function spaces, simplification of expressions, and confirmation of orthonormality in multi-dimensional analysis.