On a paper of Bojanić and Stanojević

Abstract

It is pointed out that theL ¹ convergence classes of Fourier series of even functions considered in [2] and [3] are subclasses of theL ¹ convergence classes of Fomin [7]. Moreover, showing the simple fact that for eachp>1 the classesC _p andV _p considered in [2] coincide, whereC _p={(a k):a _k=o (1) and$n^{p - 1} \sum\limits_{k = n}^\infty {\left| {\Delta _{ak} } \right|^p = o} \left( 1 \right)$}, it follows that the main result in [2] reduces to an earlier theorem of Fomin [5]. Other equivalent descriptions of these classes are also given.

Next we consider the problem of integrability of cosine series in regard to the classesC _p p>1 andC ₁=B V the set of null sequences of bounded variation. Noticing that the cosine series converges pointwise a.e. whenever its coefficients belong to ∪{C _p:p>1}, we obtain necessary and sufficient conditions for such series to be Fourier series. These results extend the second theorem in [2] and show that the classical question on integrability of cosine series need not be restricted to series with coefficients of bounded variation.