The Fourier Transform

Note. In this chapter, unless otherwise indicated, all functions are complex-valued functions of a real variable. Given integrable functions of t, f, and k the function $$g\left( \omega \right) = \int {_{E}f\left( t \right)k} \left( {t,\omega } \right)dt$$ dt for some set E is called and INTEGRAL TRANSFORM of f with KERNEL k ( t, w) (of the transform). By “transforming” both side of certain equations, we can sometimes convert them into simpler ones—differential equations to algebraic equations, for examples.