Harmonic analysis

Harmonic analysis studies how functions can be represented and analyzed through oscillatory components, such as sines, cosines, and complex exponentials.

Fourier series: A representation of a periodic function as a sum of sine and cosine waves. It decomposes periodic functions into discrete oscillatory components.
Complex exponential: A function of the form e^{ix}, whose real and imaginary parts are cosine and sine. Complex exponentials combine sine and cosine oscillations into one convenient basis.
Joseph Fourier: A French mathematician and physicist whose work on heat conduction developed Fourier series. His heat equation investigations established the method’s early importance.
Digital signal processing: The analysis and modification of signals represented as discrete numerical sequences. Fourier methods reveal frequencies and enable efficient filtering.
Representation theory: The study of mathematical structures through their actions as linear transformations on vector spaces. Group representations generalize frequency components to symmetries and structured spaces.
Fourier transform: An integral transform that expresses a function in terms of its frequency components. It extends frequency decomposition from periodic signals to nonperiodic ones.
Orthogonality: A relation in which objects have zero inner product under a specified inner product structure. Orthogonal oscillations can be separated and measured independently.
Heat equation: A partial differential equation describing the diffusion of heat through a material. Fourier used oscillatory series to solve this equation.
Fast Fourier transform: An algorithm that computes the discrete Fourier transform using fewer operations than direct evaluation. It makes frequency analysis computationally practical for large datasets.
Harmonic analysis on groups: The study of Fourier-type decompositions using the representations of a group. It extends classical analysis to settings shaped by symmetry.

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