2.1 Fourier series

Fourier series represent periodic functions as infinite sums of sine and cosine waves. This powerful tool in Approximation Theory allows complex functions to be broken down into simpler components, making analysis and manipulation easier.

Understanding Fourier series is crucial for many fields, including signal processing and engineering. We'll cover their definition, convergence properties, computation methods, and applications, as well as their relationship to other mathematical concepts.

Definition of Fourier series

• Fourier series is a fundamental concept in Approximation Theory that represents a periodic function as an infinite sum of trigonometric functions
• It allows for the approximation of complex periodic functions using a combination of simple sine and cosine waves
• Fourier series has wide-ranging applications in various fields, including signal processing, physics, and engineering

Trigonometric series representation

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• A Fourier series represents a periodic function $f(x)$ as an infinite sum of sine and cosine terms: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$
• The coefficients $a_n$ and $b_n$ determine the amplitude of each trigonometric term in the series
• The trigonometric terms have frequencies that are integer multiples of the fundamental frequency of the periodic function

Periodic functions

• Fourier series are used to represent periodic functions, which are functions that repeat their values at regular intervals
• A function $f(x)$ is periodic with period $T$ if $f(x+T) = f(x)$ for all $x$
• Examples of periodic functions include sine and cosine waves, square waves, and sawtooth waves

Coefficients of Fourier series

• The coefficients $a_n$ and $b_n$ in a Fourier series are determined by the following integrals over one period $T$:
  • $a_0 = \frac{2}{T} \int_0^T f(x) dx$
  • $a_n = \frac{2}{T} \int_0^T f(x) \cos(nx) dx$
  • $b_n = \frac{2}{T} \int_0^T f(x) \sin(nx) dx$
• These coefficients capture the contribution of each trigonometric term to the overall function approximation

Convergence of Fourier series

• The convergence of a Fourier series refers to how well the series approximates the original function as the number of terms increases
• Convergence can be analyzed in terms of pointwise convergence and uniform convergence
• Understanding the convergence properties of Fourier series is crucial for determining the accuracy of the approximation

Pointwise vs uniform convergence

• Pointwise convergence means that the Fourier series converges to the original function at each individual point
  • Formally, $\lim_{N \to \infty} S_N(x) = f(x)$ for each $x$, where $S_N(x)$ is the partial sum of the first $N$ terms of the Fourier series
• Uniform convergence is a stronger condition that requires the Fourier series to converge uniformly over the entire interval
  • The maximum difference between the partial sum $S_N(x)$ and the original function $f(x)$ approaches zero as $N$ increases

Dirichlet conditions for convergence

• The Dirichlet conditions are sufficient conditions for the pointwise convergence of a Fourier series
• The conditions require the function $f(x)$ to be:
  1. Periodic with period $2\pi$
  2. Piecewise continuous on the interval $[0, 2\pi]$
  3. Have a finite number of maxima and minima on the interval $[0, 2\pi]$
• If these conditions are met, the Fourier series of $f(x)$ will converge pointwise to $f(x)$ at all points where $f(x)$ is continuous

Gibbs phenomenon

• The Gibbs phenomenon occurs when a Fourier series approximates a function with a discontinuity
• Near the discontinuity, the partial sums of the Fourier series exhibit oscillations that overshoot and undershoot the actual function values
• The overshoots and undershoots do not disappear as the number of terms in the series increases, but their width decreases
• The Gibbs phenomenon highlights the limitations of Fourier series in approximating functions with discontinuities

Properties of Fourier series

• Fourier series possess several important properties that make them useful for analyzing and manipulating periodic functions
• These properties include linearity, shift invariance, Parseval's identity, and the convolution theorem
• Understanding these properties allows for efficient computation and manipulation of Fourier series

Linearity and shift invariance

• Linearity: If $f(x)$ and $g(x)$ have Fourier series representations with coefficients $a_n, b_n$ and $c_n, d_n$, respectively, then the Fourier series of $\alpha f(x) + \beta g(x)$ has coefficients $\alpha a_n + \beta c_n$ and $\alpha b_n + \beta d_n$
• Shift invariance: If $f(x)$ has a Fourier series with coefficients $a_n$ and $b_n$, then the Fourier series of $f(x-x_0)$ has coefficients $a_n \cos(nx_0) + b_n \sin(nx_0)$ and $b_n \cos(nx_0) - a_n \sin(nx_0)$

Parseval's identity

• Parseval's identity relates the energy of a function to the energy of its Fourier coefficients
• It states that the integral of the squared function over one period equals the sum of the squares of its Fourier coefficients: $\int_0^T |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$
• Parseval's identity is useful for analyzing the energy distribution of a function in the frequency domain

Convolution theorem

• The convolution theorem relates the Fourier series of the convolution of two periodic functions to the product of their individual Fourier series
• If $f(x)$ and $g(x)$ have Fourier series with coefficients $a_n, b_n$ and $c_n, d_n$, respectively, then the Fourier series of their convolution $f(x) * g(x)$ has coefficients $\frac{1}{2}(a_n c_n - b_n d_n)$ and $\frac{1}{2}(a_n d_n + b_n c_n)$
• The convolution theorem simplifies the computation of the Fourier series of the convolution of two functions

Computation of Fourier coefficients

• Computing the Fourier coefficients is a crucial step in constructing the Fourier series representation of a function
• The coefficients can be calculated using integration techniques, taking into account the properties of the function being approximated
• Efficient computation of Fourier coefficients is important for practical applications of Fourier series

Fourier coefficients for even vs odd functions

• Even functions: If $f(x)$ is an even function, i.e., $f(-x) = f(x)$, then its Fourier series contains only cosine terms, and the coefficients $b_n$ are zero
  • The coefficients $a_n$ can be computed using the simplified integral: $a_n = \frac{4}{T} \int_0^{T/2} f(x) \cos(nx) dx$
• Odd functions: If $f(x)$ is an odd function, i.e., $f(-x) = -f(x)$, then its Fourier series contains only sine terms, and the coefficients $a_n$ are zero
  • The coefficients $b_n$ can be computed using the simplified integral: $b_n = \frac{4}{T} \int_0^{T/2} f(x) \sin(nx) dx$

Orthogonality of trigonometric functions

• The trigonometric functions used in Fourier series, i.e., sine and cosine functions with integer multiples of the fundamental frequency, form an orthogonal set over the interval $[0, T]$
• Orthogonality means that the integral of the product of any two different trigonometric functions over one period is zero:
  • $\int_0^T \cos(nx) \cos(mx) dx = 0$ for $n \neq m$
  • $\int_0^T \sin(nx) \sin(mx) dx = 0$ for $n \neq m$
  • $\int_0^T \cos(nx) \sin(mx) dx = 0$ for all $n$ and $m$
• The orthogonality property simplifies the computation of Fourier coefficients and allows for the unique representation of a function by its Fourier series

Integration techniques for coefficient calculation

• Computing Fourier coefficients involves evaluating integrals of the form $\int_0^T f(x) \cos(nx) dx$ and $\int_0^T f(x) \sin(nx) dx$
• Various integration techniques can be employed depending on the nature of the function $f(x)$:
  • Analytical integration: If $f(x)$ has a closed-form antiderivative, the integrals can be evaluated directly
  • Numerical integration: If $f(x)$ is known only at discrete points or is difficult to integrate analytically, numerical methods such as the trapezoidal rule or Simpson's rule can be used
  • Symmetry properties: Exploiting the even or odd symmetry of $f(x)$ can simplify the integration process, as shown in the previous section

Applications of Fourier series

• Fourier series have numerous applications in various fields, including mathematics, physics, engineering, and signal processing
• They provide a powerful tool for analyzing and manipulating periodic functions and signals
• Some key applications of Fourier series include approximation of functions, solving differential equations, and signal processing and filtering

Approximation of functions

• Fourier series can be used to approximate periodic functions by truncating the series to a finite number of terms
• The more terms included in the truncated series, the better the approximation to the original function
• Fourier series approximations are useful for representing complex functions in a simpler form, which can facilitate analysis and computation

Solving differential equations

• Fourier series can be employed to solve certain types of differential equations, particularly those involving periodic boundary conditions
• The general approach involves assuming a Fourier series solution, substituting it into the differential equation, and solving for the Fourier coefficients
• This technique is commonly used in heat transfer, wave propagation, and vibration analysis problems

Signal processing and filtering

• In signal processing, Fourier series are used to represent periodic signals as a sum of sinusoidal components
• The Fourier coefficients provide information about the frequency content of the signal
• Filtering operations can be performed by manipulating the Fourier coefficients to remove or attenuate certain frequency components
• Examples include low-pass, high-pass, and band-pass filters, which are used to remove noise or extract specific frequency ranges from a signal

Generalized Fourier series

• The concept of Fourier series can be extended and generalized to accommodate a wider range of functions and domains
• Generalized Fourier series include complex Fourier series, Fourier series on arbitrary intervals, and Fourier series for non-periodic functions
• These generalizations expand the applicability of Fourier series to a broader class of problems

Complex Fourier series

• Complex Fourier series represent a periodic function using complex exponential functions instead of sine and cosine functions
• The complex Fourier series of a function $f(x)$ with period $T$ is given by: $f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i\frac{2\pi n}{T}x}$ where $c_n$ are the complex Fourier coefficients, computed using the integral: $c_n = \frac{1}{T} \int_0^T f(x) e^{-i\frac{2\pi n}{T}x} dx$
• Complex Fourier series provide a more compact representation and simplify certain mathematical operations, such as differentiation and integration

Fourier series on arbitrary intervals

• Fourier series can be defined on arbitrary intervals $[a, b]$ instead of the standard interval $[0, T]$
• The Fourier series of a function $f(x)$ on the interval $[a, b]$ is given by: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{n\pi}{b-a}(x-a)\right) + b_n \sin\left(\frac{n\pi}{b-a}(x-a)\right)\right)$ where the coefficients $a_n$ and $b_n$ are computed using the integrals:
  • $a_0 = \frac{2}{b-a} \int_a^b f(x) dx$
  • $a_n = \frac{2}{b-a} \int_a^b f(x) \cos\left(\frac{n\pi}{b-a}(x-a)\right) dx$
  • $b_n = \frac{2}{b-a} \int_a^b f(x) \sin\left(\frac{n\pi}{b-a}(x-a)\right) dx$
• Fourier series on arbitrary intervals allow for the representation of functions defined on domains other than the standard period $[0, T]$

Fourier series for non-periodic functions

• Fourier series can be used to represent non-periodic functions by extending the function periodically outside its original domain
• The periodic extension of a non-periodic function $f(x)$ defined on the interval $[a, b]$ is given by: f(x), & a \leq x < b \\ f(x + (b-a)), & x < a \\ f(x - (b-a)), & x \geq b \end{cases}$$
• The Fourier series of the periodic extension $f_p(x)$ can then be computed using the standard Fourier series formulas
• This approach allows for the approximation of non-periodic functions using Fourier series, albeit with some limitations due to the artificial periodicity introduced by the extension

Relationship to other concepts

• Fourier series are closely related to other important concepts in mathematics and signal processing
• Understanding the connections between Fourier series and these concepts provides a deeper understanding of their properties and applications
• Some key related concepts include Fourier transforms, Taylor series, and harmonic analysis

Connection to Fourier transforms

• The Fourier transform is a generalization of the Fourier series that extends the concept to non-periodic functions
• While Fourier series represent a function as a sum of discrete frequency components, the Fourier transform represents a function as a continuous spectrum of frequencies
• The Fourier transform of a function $f(x)$ is defined as: $F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$
• The Fourier transform and Fourier series are related through the Discrete Fourier Transform (DFT), which is a numerical approximation of the Fourier transform for discrete-time signals

Comparison with Taylor series

• Taylor series and Fourier series are both used to represent functions as infinite series, but they differ in their basis functions and domains of convergence
• Taylor series represent a function as a power series around a specific point, using polynomials as basis functions: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ where $f^{(n)}(a)$ denotes the $n$-th derivative of $f$ at point $a$
• Fourier series, on the other hand, use trigonometric functions as basis functions and are particularly well-suited for representing periodic functions
• While Taylor series provide local approximations around a point, Fourier series offer global approximations over the entire period of the function

Role in harmonic analysis

• Harmonic analysis is a branch of mathematics that studies the representation of functions and signals as a superposition of basic waves or harmonics
• Fourier series play a central role in harmonic analysis, as they provide a way to decompose periodic functions into their constituent frequency components
• The study of Fourier series and their convergence properties forms the foundation for more advanced topics in harmonic analysis, such as Fourier transforms, wavelets, and time-frequency analysis
• Harmonic analysis has applications in various fields, including signal processing, quantum mechanics, and partial differential equations, where the understanding of Fourier series is crucial for analyzing and manipulating functions and signals