ABSTRACT
In article 1.3 of chapter 1 it is noted that if ∫ - ∞ + ∞ | f ( t ) | d t $ \int_{ - \infty }^{ + \infty } |f(t)|dt $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429503580/e8c17e8d-aa2b-4352-a3e7-8cf96212c2f8/content/inline-math3_1.tif"/> is not convergent, Fourier transform F(ξ) of the function f(t) need not exist for all real ξ. For example, when f ( t ) = sin ω t , $ f(t) = {\text{sin }}\omega t, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429503580/e8c17e8d-aa2b-4352-a3e7-8cf96212c2f8/content/inline-math3_2.tif"/> ω = real, F(ξ) does not exist. But such situations do arise occasionally in practice. To handle this situation, we consider a new function f 1(t) connected to f(t) defined by
