Fourier, Transform & Operator Methods

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Abstractions about representing signals and functions through convolution, Fourier and Z transforms, spectral analysis, fractional differentiation, operators, response functions, and smoothness. They include computational algorithms, special functions, dispersion relations, and multidimensional systems.

19 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bruun's FFT algorithm — A fast Fourier transform based on recursive real-coefficient factorization of the transform polynomial, postponing complex arithmetic until a final reconstruction stage.
• Circular convolution — A convolution on periodic sequences in which indices wrap modulo a fixed length, corresponding to pointwise multiplication under the discrete Fourier transform.
• Differintegral — A fractional-calculus operator D^q that unifies differentiation for positive order and integration for negative order, with integer cases recovered under a specified convention.
• Digamma function — The logarithmic derivative ψ(z)=Γ′(z)/Γ(z) of the gamma function, extending shifted harmonic-number relations to complex arguments.
• Dissipation factor — Express oscillatory loss as the ratio of dissipative to reactive response, equivalently the reciprocal of quality factor under a declared convention.
• Fourier analysis — The representation and study of functions or signals through sinusoidal or character components indexed by frequency.
• Hermitian function — A complex-valued function satisfying conjugate symmetry f(-x)=conjugate(f(x)), equivalently having an even real part and an odd imaginary part.
• Hilbert spectral analysis — A time–frequency analysis that forms analytic signals, often from empirical mode components, and derives instantaneous amplitude and frequency from Hilbert phase.
• Kramers–Kronig relations — Hilbert-transform relations connecting real and imaginary parts of a causal linear response function through analyticity in the upper complex-frequency half-plane.
• Multidimensional system — A mathematical system whose signals or states evolve over two or more independent variables, such as spatial coordinates as well as time.
• Phase response — The frequency-dependent phase difference between a system's sinusoidal output and input, forming the angular component of its complex transfer function.
• Progressive function — An L2 signal whose Fourier transform is supported only on nonnegative frequencies, equivalently a boundary function in the upper-half-plane Hardy space under the stated convention.
• Pseudo-differential operator — An operator defined through a position- and frequency-dependent symbol, extending differential operators to non-polynomial multipliers and supporting microlocal analysis of PDE.
• Random number generation — Produce symbols intended to be unpredictable or distributionally random by sampling physical entropy or evolving a deterministic pseudorandom state under an explicit seeding and output convention.
• Rectangular function — A piecewise-constant pulse equal to one on a centered finite interval and zero outside, with a convention-dependent value at the endpoints.
• Smoothness (probability theory) — Classify an error distribution by the asymptotic decay of its characteristic function, separating polynomially ordinary-smooth laws from exponentially supersmooth laws in nonparametric inverse problems.
• Stieltjes transformation — Map a measure to an analytic function off its support by integrating the resolvent kernel 1/(t−z), with boundary limits recovering density and encoding moments and spectral information.
• Swish function — A smooth neural-network activation family fβ(x)=x·sigmoid(βx) that interpolates between a scaled linear map and a ReLU-like gate while remaining mildly nonmonotonic for positive β.
• Z-transform — Represent a discrete-time sequence by a Laurent series in a complex variable together with its region of convergence, enabling shifts, convolution, recurrences, spectra, and system behavior to be analyzed algebraically.