Pointwise behavior of Fourier integrals of functions of bounded variation over \(\mathbb R\)
Let \(S_T(f,x)=\int_{-T}^T \hat f(t) e^{ixt}dt,\) \(T>0,\) where \(\hat f\) is the Fourier transform of \(f.\) The author proves that if \(f\) is integrable and of bounded variation, written \(V(f)<\infty,\) on \(\mathbb R,\) and \(\sum_{j=m}^\infty u_j^{-1}\leq Au_m^{-1},\) \(m=1,2,\dots,\) for an increasing sequence \(\{u_j\}\) of positive numbers and a constant \(A,\) then \[ \sum_{j=1}^\infty \max_{u_{j-1}\leq \nu\leq u_j}| s_\nu(f,x)-s_{u_{j-1}} (f,x)| \leq{3A+4\over\pi}V(f), \quad x\in\mathbb R, \] with \(u_0=0.\) The series converges uniformly at every point of continuity of \(f\) or even on the whole closed interval of continuity. This implies various generalizations of results, known for Fourier series, to convergence of \(S_T(f,x)\) as well as a stronger version of the nonperiodic Parseval formula. The obtained main results are, in turn, generalizations and slight refinement of the periodic prototypes proved by S.A. Telyakovskij.
scientific article; zbMATH DE number 2107205