A generalization of the Fourier cosine series
Joseph L. Walsh · Transactions of the American Mathematical Society · 1921
* Presented to the Society, September, 1920. t This study was undertaken at the suggestion of Professor Birkhoff, who proposed an analogous generalization of the Fourier sine series. There is no analogous generalization of the sine series such that developments in terms of the two sets of functions {sin nx I and {sin Xn X I , where Xn is near but not equal to n, will have the same convergence properties on the interval 0 _ x e ,r. For at the right-hand end of the interval any particular function sin Xk x is not zero and hence cannot be developed in a uniformly convergent series in terms of {sin nx I, but it can be developed in a uniformly convergent series in terms of I sin X,, x t . Thus the methods of the present paper cannot be used to give a generalization of the sine series on the interval 0 _ x 7r, but these methods with slight modifications would enable us to replace the set {sin nx I by a set {sin Xn X I such that developments in terms of the two sets of functions will have the same convergence properties on the interval 0 _ x _ , where e is arbitrary. The methods used in this paper are in spirit closely related to the methods used by Birkhoff, P a r i s C o m p t e s R e n d u s, vol. 164 (1917), pp. 942-945, who gives a similar generalization of Taylor's series. Birkhoff makes use of the theory of integral equations, but the related theory of infinitely many variables as used in the present paper leads to more general results.