PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis → Harmonic Analysis ↓


Fourier Series


Fourier series is a way of representing a function as a sum of sine and cosine functions. The method is named after the French mathematician Jean-Baptiste Joseph Fourier, who introduced the concept of Fourier series in the study of heat transfer. In short, Fourier series allows us to express a complex periodic function as a sum of simple oscillating functions.

Introduction to fourier series

When dealing with periodic functions such as oscillations or waves, Fourier series come into play by providing a technique to decompose such periodic functions into a set of simpler sine and cosine functions. These simpler functions are called "harmonics" and are important in understanding complex oscillatory behavior.

f(x) = a_0 + a_1*cos(x) + b_1*sin(x) + a_2*cos(2x) + b_2*sin(2x) + ...

In this equation, a_0, a_n, and b_n are coefficients that are determined based on the function f(x) we want to represent. Each cosine and sine term corresponds to a harmonic of the fundamental frequency.

Mathematical formulation

Given a function f(x) with period 2π, it can be expressed by a Fourier series as follows:

f(x) = frac{a_0}{2} + sum_{n=1}^{infty} left( a_n cos(nx) + b_n sin(nx) right)

where the coefficients a_n and b_n are calculated as follows:

a_0 = frac{1}{pi} int_{-pi}^{pi} f(x) , dx a_n = frac{1}{pi} int_{-pi}^{pi} f(x) cos(nx) , dx b_n = frac{1}{pi} int_{-pi}^{pi} f(x) sin(nx) , dx

These formulas allow us to calculate the appropriate weights of each harmonic that will form the original function using a Fourier series.

Visual example

To understand how Fourier series work, consider the SVG representation of a function and its Fourier approximation:

Basic function (blue) Fourier approximation (red)

In this SVG example, the blue curve represents the original harmonic function, and the red line represents the Fourier approximation using a finite number of terms. Even with a few harmonics, the approximation can be quite accurate.

Examples and applications

Fourier series are ubiquitous in various fields such as physics, engineering, and signal processing. Here are some practical examples:

  • Signal processing: Fourier series are used to analyze sound waves, electrical signals, and other types of waveform data.
  • Electronic engineering: In designing circuits, Fourier series helps in understanding and creating filters and signal modulation schemes.
  • Heat transfer: Fourier introduced this series in his analytical solution to the one-dimensional heat equation.

Consider a practical application involving sound waves. Sound is essentially a wave and can be represented as a function of time, say s(t) Using a Fourier series, this complex wave can be decomposed into simple sine and cosine waves, each of which represents different frequencies:

s(t) = frac{a_0}{2} + a_1 cos(w_1 t) + b_1 sin(w_1 t) + a_2 cos(w_2 t) + b_2 sin(w_2 t) + ...

where w_n represents the angular frequency of the harmonics. This decomposition allows audio engineers to analyze and modify sound recordings by working with individual harmonics.

Convergence of Fourier series

The convergence of a Fourier series to a real function depends on certain conditions. If f(x) is a periodic and continuous function, and has a finite number of minima and maxima, then the Fourier series will converge to f(x) at every point where f is continuous. At the points of discontinuity, the series will converge to the average of the left and right limits.

This feature of the Fourier series is often described by the Dirichlet conditions:

  • The function f(x) must be 2π-periodic.
  • f(x) must be piecewise smooth within each period.

The convergence of Fourier series plays an important role in the modeling of phenomena where instantaneous changes are approximated by smooth transitions.

Summary and conclusion

Fourier series are one of the most powerful tools in mathematical analysis and applied mathematics, because of their ability to transform complex periodic functions into compositions of simple sines and cosines. Their application spans across science and engineering, providing insights into wave behavior, signal processing, structural analysis, and many other areas.

By understanding the beauty of Fourier series, researchers can decode complex systems into fundamental waveforms, simplify analysis, and enable important innovations. Fourier's insights into heat transfer calculations have resonated across fields, demonstrating the continuing relevance of mathematical exploration in the applied sciences.

As this lesson demonstrates, much of the complexity of natural phenomena can be simplified by accepting and using the principles of Fourier series. Whether in theory or in practical applications, the combination of simplicity and power in this method is unmatched.


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Fourier Series

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