Determination of a jump by conjugate Fourier-Jacobi series

Abstract

In the present paper we prove equiconvergence theorem of conjugate Fourier-Jacobi series and the dierentiated Fourier-Jacobi series. Next,
based on this theorem we establish Cesaro summability result of conjugate
Fourier-Jacobi series for
1
2
;
1
2
; of a function of bounded pvariation, p > 1; and accordingly obtain new ways for determination of jump
discontinuities. The classical theorem of Ferenc Lukacs [11] determines the
generalized jumps of a periodic, Lebesgue integrable function f in terms of
the partial sum of the conjugate series to the Fourier series of f: We prove
an analogous theorem in terms of the partial sum of conjugate Fourier-Jacobi
series, for
1
2
;
1
2
: