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Integral Transformation 📂Linear Algebra

Integral Transformation

Definition

If a map JJ from a function space to a function space is defined as the following integral, then JJ is called an integral transform.

(Jf)(x)=∫abK(x,t)f(t)dt (Jf) (x) = \int_{a}^{b} K(x,t)f(t)dt

J:f(⋅)↦∫abK(⋅,t)f(t)dt J : f(\cdot) \mapsto \int_{a}^{b} K(\cdot,t)f(t)dt

In this case, KK is referred to as the kernel of JJ. If a map from JfJf to ff exists, it is denoted as J−1J^{-1} and called the inverse transform of JJ.

Description

Since integration is linear, integral transforms are linear transforms.

The integration domain does not necessarily have to be bounded. It can be a=−∞a=-\infty, b=∞b=\infty, or both. Although integral transforms can be created arbitrarily according to the definition, to have meaningful interpretations, solving the given problem should be easier in terms of JfJf than in ff, or an inverse transformation should exist, allowing free conversion between JfJf and ff. Examples of integral transforms include the following.

  • Fourier transform F\mathcal{F}:

    Ff(ξ)=∫−∞∞f(x)eiξxdx,K(x,ξ)=eiξx \mathcal{F}f(\xi)=\int _{-\infty} ^{\infty} f(x)e^{i \xi x}dx,\quad K(x,\xi)=e^{i\xi x}

  • Laplace transform L\mathcal{L}:

    Lf(s)=∫0∞f(t)e−stdt,K(t,s)=e−st \mathcal{L}f(s)=\int _{0} ^{\infty}f(t)e^{-st}dt,\quad K(t,s)=e^{-st}

  • Mellin transform M\mathcal{M}:

    Mf(s)=∫0∞f(x)xs−1dx,K(x,s)=xs−1 \mathcal{M}f(s)=\int_{0}^{\infty} f(x)x^{s-1}dx,\quad K(x,s)=x^{s-1}

  • Radon transform R\mathcal{R}:

    Rf(s,θ)=∫−∞∞f(scos⁡θ−tsin⁡θ,ssin⁡θ+tcos⁡θ)dt \mathcal{R}f(s,\theta)=\int_{-\infty}^{\infty}f(s\cos\theta-t\sin\theta, s\sin\theta+t\cos\theta)dt

See Also