Fourier analysis

Fourier analysis represents functions or signals as sums or integrals of sinusoidal components. Its central tools, Fourier series and Fourier transforms, reveal how their frequency components are distributed.

Fourier series: A representation of a periodic function as a sum of sine and cosine terms, or complex exponentials. It decomposes periodic functions into discrete harmonics.
Complex exponential: A function of the form e to a complex-valued argument, whose oscillatory behavior encodes sine and cosine. Complex exponentials provide compact notation for sinusoidal components.
Digital signal processing: The analysis and manipulation of signals represented as discrete numerical samples. Fourier methods underpin filtering, spectral estimation, and many digital signal operations.
Wavelet transform: A transform that represents signals with localized waveforms at multiple scales and positions. Unlike global sinusoids, wavelets retain information about where signal features occur.
Jean-Baptiste Joseph Fourier: French mathematician and physicist who developed heat theory and advanced trigonometric-series methods. His work on heat conduction made series representations central to mathematical physics.
Fourier transform: An integral transform that expresses a function in terms of its continuous frequency components. It extends frequency decomposition to nonperiodic functions.
Orthogonality: A relation in which objects have zero inner product, generalizing perpendicularity to functions and vectors. Orthogonal basis functions let Fourier coefficients isolate individual components.
Image compression: The reduction of image data by encoding visual information with fewer bits. Transform coefficients concentrate image information for compact encoding.
Laplace transform: An integral transform that represents functions using complex exponential kernels and often handles growth or decay. Its broader convergence behavior suits differential equations where Fourier integrals may fail.
Heat equation: A partial differential equation describing how temperature diffuses through a material over time. Fourier developed his series methods to solve problems of heat propagation.

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