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


Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as superpositions of fundamental waves, and the study and generalization of Fourier series and Fourier transforms. It can be seen as an exploration of how functions can be broken down into simpler, "harmonic" components.

Introduction

To understand harmonic analysis, let's consider a simple example: Imagine you are listening to your favorite song. What you are actually hearing is a complex mixture of different sounds that reach your ears as a single wave. However, this wave can be decomposed into many simpler waves, each of which has a different frequency, amplitude, and phase. This decomposition is a form of harmonic analysis.

Fourier Series

The Fourier series is a way of representing a periodic function as a sum of simple sine and cosine waves. It is the starting point of harmonic analysis.

The periodic function f(x) with period 2π can be expressed as:

f(x) = a_0/2 + Σ [a_n cos(nx) + b_n sin(nx)], (n=1 to ∞)

Here, a_n and b_n coefficients are calculated as:

a_n = 1/π ∫ from 0 to 2π of f(x)cos(nx) dx
b_n = 1/π ∫ from 0 to 2π of f(x)sin(nx) dx
sine wave

Visualization of Fourier Series

Suppose we have a simple square wave. To decompose it using a Fourier series, we have a series of sine functions with different frequencies that add up to form a square wave.

square wave

Harmonic Series Example

Sometimes, you can look at a wave as a superposition of its harmonics rather than as separate sine and cosine functions. Its simplest form is:

f(x) = sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + (1/7)sin(7x) + ...

This series shows how complex signals can be built from simple, individual frequencies (harmonics).

Applications of Harmonic Analysis

Signal processing

Harmonic analysis plays an important role in the field of signal processing. Modern electronics use Fourier transforms to decode signals, compress audio and video files, and much more.

Music and acoustics

In music, harmonic analysis helps to understand how different sound waves combine to create harmony. Musicians use this analysis when tuning instruments.

Quantum mechanics

Some concepts in quantum mechanics, such as the wave function, use harmonic analysis to represent states and study particle behavior.

Fourier Transform

The Fourier transform is a mathematical concept that converts a function of time into a function of frequency.

F(k) = ∫ from -∞ to +∞ of f(x)e^(-2πixk) dx

The resulting function represents the frequency spectrum of the original function.

Frequency spectrum

Convolution Theorem

In harmonic analysis, the convolution theorem is important for simplifying the problem of finding the convolution of a function. In simple terms:

Fourier Transform of (f * g) = Fourier Transform of f × Fourier Transform of g

This duality between convolution in the time domain and multiplication in the frequency domain simplifies many calculations in engineering and physics.

Wavelets

Wavelets are modern tools of harmonic analysis that help analyze data with abrupt changes. Unlike Fourier transforms, wavelets are better suited for time-varying signals.

The continuous wavelet transform is:

C(a, b) = ∫ from -∞ to +∞ of f(x)ψ((xb)/a) dx

Here, ψ(x) is the wavelet function, a is the scale, and b is the position.

Conclusion

Harmonic analysis is integral to understanding the mathematical basis of various phenomena in science and engineering. From the music we listen to to the technology we use in our daily lives, harmonic analysis helps us understand complex signals by breaking them down into their basic components.

Whether using Fourier series for periodic functions or wavelets for changing signals, the insights provided by harmonic analysis are invaluable in many fields. Advances in harmonic analysis continue to uncover new methods and applications, proving its continued relevance and potential.


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Harmonic Analysis

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