Transformations of Fourier coefficients
Frederick H. Young · Proceedings of the American Mathematical Society · 1952
Introduction. In 1923 M. Fekete [2]2 introduced the concept of factor sequences that left invariant the class of a Fourier series. That is, Fekete investigated the conditions to which a sequence of constants (X„) must be subjected in order that (X„a„, X„i„) be Fourier coefficients of a function of the same class, K, as that of the function determined by (an, bn). Whenever (X„) has this property, (X„) is said to belong to the class (£, K). Fekete restricted his investigation to those cases for which K represented the class of continuous, essentially bounded, Riemann integrable, or Lebesgue integrable functions, functions of bounded variation, or functions having a LebesgueStieltjes series. It may be mentioned in passing that Verblunsky [5] extended Fekete's results to some of the cross classes. Note that the transformation effected by a factor sequence may be accomplished by matric multiplication. Suppose that the cosine coefficients of/ are (an). Then the coefficients of the function resulting from the transformation appear in the column vector on the right below.