Type: Article
Publication Date: 1996-06-01
Citations: 3
DOI: https://doi.org/10.1017/s1446788700037903
Abstract We study cosine and sine Fourier transforms defined by F(t) := (2/π) and ( t ):= (2/π), where f is L 1 -integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are ( L 1 -integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras S ν ( f, x ):= and ν (f, x) := , the modified partial integrals u ν (f, x) := s ν (f, x) - F (ν)(sin νx)/x and ũ ν ( f, x ):= ν ( f, x ) + (ν) (cos ν x )/ x , where ν > 0. We give necessary and sufficient conditions for( L 1 [0, ∞)-convergence of u ν ( f ) and ũ ν ( f ) as well as for the L 1 [0, X]-convergence of s ν ( f ) and ν ( f ) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that s ν ( f ) and ν ( f ) cannot belong to ( L 1 [0,∞). Conequently, it makes no sense to speak of their ( L 1 [0, ∞)-convergence as ν ← ∞. As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind. We extend these results to the complex Fourier transform defined by G(t) := , where g is L 1 - integrable over (−∞, ∞).