On the Fourier integral theorem

Introduction. In traditional proofs of convergence of Fourier series and of the Fourier integraI theorem basic tools are the theory of Dirichlet integraIs and the Riemann-Lebesgue lemma. Recently CHERNOFF [I) and REoIlEFFER (2) gave new proofs of convergenceof Fourier series which make no use of the Dirichlet theory. Moreover, they need only the special case of the Riemann-Lebesgue lemma which states that the Fourier coeflicients of a Lebesgue integrabIe function tend to zero. Further, the authors consider asymmetrie partial sums of the (complex) Fourier series instead of the usuaI symmetrie partial sums. The purpose of this note is to prove the Fourier integral theorem in an analogous manner.