Determination of the Jumps of a Bounded Function by Its Fourier Series

AbstractThe well-known identity which determines the jumps of a function of bounded variation by its Fourier series is extended to larger classes of functions, such asVΦ,ΛBV, andV[v], under some conditions on the generalized variations. It is shown as well that the conditions on the generalized variations are definitive in some sense. Based on the above-mentioned results, an identity which determines the jumps of a bounded function by its Fourier series with respect to the system of generalized Jacobi polynomials is obtained for these function classes