Fourier Series

AT the thirty-second session of the Indian Science Congress, held at Nagpur in 1945, Dr. B. N. Prasad, president of the Section of Mathematics and Statistics, delivered an address on the summability of a Fourier series and its conjugate series. Fourier series are of very great importance in both pure mathematics and mathematical physics. Fourier himself dealt with physical applications, ignoring theoretical difficulties, but these difficulties have proved of great interest. It has long been known that the Fourier series corresponding to a continuous function may fail to give the correct values of this function at an infinite set of points. To get over this difficulty, we may use a generalization of the process of finding the sum of an infinite series, such as that due to Cesaro. It may be recalled that Hardy and Rogosinski, in their Cambridge Tract on Fourier series, could not find space for theorems concerning Cesaro summation of general order. Thus Dr. Prasad's address, which is clearly written, is a useful supplement to the Tract. Dr. Prasad himself has made many contributions to the subject.