DETERMINATION OF THE JUMPS OF A BOUNDED-FUNCTION BY ITS FOURIER-SERIES

The well-known identity which determines the jumps of a function of bo unded variation by its Fourier series is extended to larger classes of functions, such as V-phi, Lambda BV, and V[upsilon], under some condi tions on the generalized variations. It is shown as well that the cond itions on the generalized variations are definitive in some sense. Bas ed on the above-mentioned results, an identity which determines the ju mps of a bounded function by its Fourier series with respect to the sy stem of generalized Jacobi polynomials is obtained for these function classes. (C) 1998 Academic Press.