An integral transform constructs a new function by integrating an input against a kernel:The kernel specifies how each input location contributes to an output indexed by . Whenever the integrals exist, linearity gives . The function spaces, convergence conditions and normalization are part of the definition; a formal kernel alone need not define an invertible or even everywhere-defined linear operator.[ref]
The Laplace transform uses on nonnegative time. For it gives when . The Fourier transform decomposes a function into oscillatory frequencies, while the Mellin transform is adapted to changes of scale. A smoothing Gaussian kernel instead gives the Weierstrass transform.
The useful operation is often a change of representation: the Laplace transform of a derivative turns a constant-coefficient differential equation into an algebraic equation with initial data. The inverse step then requires a theorem, such as the Bromwich inversion formula, with its own regularity and convergence assumptions.
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