Integral_transform - Wikiparency

# Integral transform  ## Microsims — p5.js ### Integral transform (p5.js) · `∫ K(t,s)x dt` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/WzsFJGbDx" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Integral transform — p5.js microsim"></iframe> </div> *The general template — Fourier, Laplace, and friends are all integrals against a kernel.* **Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/WzsFJGbDx) · movement *II · Transforms & the frequency domain* · library `p5js` ### Related microsims Live sims on neighbouring articles — 5 of them inside this article's own Wikipedia link tree: - [[Calculus]] *(in tree)* - [[Fourier_analysis]] *(in tree)* - [[Frequency_domain]] *(in tree)* - [[Laplace_transform]] *(in tree)* - [[Time_domain]] *(in tree)* - [[Complex_analysis]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## Links (Wikipedia order)  `*-algebra` · `Abel's_test` · `Abel_transform` · `Almost_everywhere` · `Alternating_series` · `Alternating_series_test` · `Antiderivative` · `Babenko–Beckner_inequality` · `Banach_space` · `Bateman_transform` · `Bessel's_inequality` · `Binomial_series` · `Bochner_space` · `Bounded_function` · `Brunn–Minkowski_theorem` · `C*-algebra` · [[Calculus]] · [[Calculus_of_variations]] · `Calculus_on_Euclidean_space` · `Cauchy_condensation_test` · `Cauchy–Schwarz_inequality` · `Chain_rule` · `Characteristic_equation_(calculus)` · `Chebyshev's_inequality` · `Chebyshev_distance` · `Circulant_matrix` · `Circular_convolution` · `Clarkson's_inequalities` · `Compact_operator` · `Continuous_function` · `Contour_integration` · `Convergence_in_measure` · `Convergence_tests` · `Curl_(mathematics)` · `Cyclic_group` · `Derivative` · `Differential_(mathematics)` · `Differential_calculus` · [[Differential_equation]] · 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`Order_of_integration_(calculus)` · `Orthonormal_basis` · `Parseval's_identity` · `Partial_derivative` · `Plancherel_theorem` · `Poisson_kernel` · `Polarization_identity` · `Power_rule` · `Power_series` · `Precalculus` · `Probability_theory` · `Product_rule` · `Propagator` · `Prékopa–Leindler_inequality` · `Pythagorean_theorem` · `Quasinorm` · `Quotient_rule` · `Radon_transform` · `Ratio_test` · `Real_analysis` · `Related_rates` · `Reynolds_transport_theorem` · `Riemann_integral` · `Riemann–Lebesgue_lemma` · `Riesz–Fischer_theorem` · `Riesz–Thorin_theorem` · `Risch_algorithm` · `Rolle's_theorem` · `Root_test` · `Schwartz_kernel_theorem` · `Second_derivative` · `Sequence_space` · `Series_(mathematics)` · `Shell_integration` · `Sobolev_space` · `Square-integrable_function` · `Stochastic_calculus` · `Stochastic_discount_factor` · `Stokes'_theorem` · `Surface_integral` · `Symbolic_integration` · `Symmetric_decreasing_rearrangement` · `Symmetry_of_second_derivatives` · `Tangent_half-angle_substitution` · `Taxicab_geometry` · `Taylor's_theorem` · `Taylor_series` · [[Time_domain]] · `Transformation_(function)` · `Trigonometric_substitution` · `Two-sided_Laplace_transform` · `Uniform_norm` · `Variable_(mathematics)` · `Vector_calculus` · `Vector_calculus_identities` · `Vitale's_random_Brunn–Minkowski_inequality` · [[Voltage]] · `Volume_integral` · `Von_Neumann_algebra` · `Weierstrass_transform` · `X-ray_transform` · `Young's_convolution_inequality` > Signal Processing concept · part of the Signal Processing Portal · movement II · !31 変換 henkan.svg  !480 *Rendered from the live microsim (▶ motion).*  ## See it next [![Fourier transform|200](Fourier_transform_thumb.png)](Fourier_transform) *→ Fourier transform*  --- Back to Signal Processing Portal · the room · Semiotic gateway ## What it is An integral transform maps a function to a new function by integrating it against a fixed kernel K(t,s), i.e. F(s) = ∫ K(t,s) f(t) dt over some range. ## How it works / why it matters The choice of kernel defines the transform: the Fourier transform uses e^{−jωt}, the Laplace transform uses e^{−st}, and there are Mellin, Hankel, and wavelet kernels too. Such transforms convert differentiation and convolution into simpler algebraic operations, so a hard problem in one domain often becomes an easy one in the transform domain, to be inverted afterward. They form the unifying mathematical backbone of signal analysis and the solution of differential equations. ## Signs & universals Instantiates: transformation, signal, frequency. ## Related Fourier transform · [[Laplace_transform]] · Linear canonical transformation · Wavelet transform · Transform theory · [[Z-transform]]   *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Integral_transform) : [Wikitube](https://en.wikitube.io/wiki/Integral_transform) ## Previous hub tags Tree parent: [[Phase_space]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*