Some theorems on Fourier transforms and conjugate trigonometric integrals

sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Réelle, 1926, p. 102.1936] FOURIER TRANSFORMS 289 R | »(*) | + [if'ix))2 + (/'(x))2]"2 ^ MR.Zygmund showed that if <#>(») is a non-decreasing convex function of m(m ^0), then r2t /I/'(*)I\ f ** , i I 4>( Jdx^ I «(|/(*) | )d* with the same inequality for <p('(x)\/R).These are the results of Szegö and Zygmund which wé generalize.We show that if /(*) has the form (2), with a(t)-a( -/),j and if |/(x)| gtf, -co <x < oo, then (3) holds, where <t(x) = j (í-^**'da(t); and that Zygmund's result may be generalized to the inequality /" /1 cos e _ sin e \ r * , *(U*)+-/'(*)+-A*) )dx^J *(|/W|)d*, where <£(») is non-negative, non-decreasing, and convex.From ( 3) we may also obtain the generalization to integrals of a result of van der Corput and Schaake,{ namely, that under the same hypotheses onIn §5 we apply our results to obtain inequalities for conjugate trigonometric integrals.We can generalize to trigonometric integrals a theorem of M. Fekete, § which states that if /(*) is measurable on (-x, tt) and periodic with the period 2ir; if |/(x) | ^ M ; and if S"(x) is the sum of the first « terms of the series conjugate to the Fourier series of/(*), then | s"(x)| ^CM log », where C is an absolute constant.We shall show that ii fix) has the form (2), with «(<)=«( -t), and ail) constant on 0</<l, then |/(x)| ^M, -oo