transformata

Transformata, in mathematics, is a broad term for a mapping or operator that takes an input object such as a function or sequence to an output object, often in a form that reveals useful structure. Transforms typically convert data into a different domain, for example from time to frequency or from space to a coefficient representation, thereby simplifying analysis, solving equations, or revealing patterns. Many transforms are linear and invertible, and several satisfy the convolution theorem, turning convolution in the original domain into multiplication in the transformed domain.

Two broad categories are integral transforms and discrete transforms. Integral transforms define a new function F(s) = ∫ K(s,t) f(t) dt using a kernel K, with an inverse transform recovering f from F. Discrete transforms operate on finite or countable sequences, such as the discrete Fourier transform (DFT) or discrete cosine transform (DCT).

Classic examples include the Fourier transform, which maps a function to its frequency spectrum; the Laplace transform, used for solving differential equations; the Hilbert transform, related to phase shifts; and the Z-transform, central in digital signal processing. Wavelet transforms provide time–frequency localization, and other transforms such as Mellin and Hankel have specialized uses.

Applications span signal processing, image and audio compression, solving differential equations, data analysis, and quantum mechanics. The concept originated with Fourier’s work on heat conduction, with later development of Laplace transforms and discrete transforms to support computation.

See also: Fourier transform, Laplace transform, Z-transform, Wavelet transform, Hilbert transform.