Spectral Rigidity for the Hilbert Transform and Related Piecewise Fourier Multipliers

We develop a unified setting for rigidity phenomena in harmonic analysis associated with Fourier integral transforms defined by discontinuous multipliers. We show that jump discontinuities in the spectral symbol induce strong incompatibilities between spatial and spectral localization properties. In particular, we establish that for a broad class of Fourier multiplier operators, unilateral spectral support combined with local vanishing of the transformed function forces triviality. This extends classical Titchmarsh-type results beyond convolution structures and highlights the fundamental role of discontinuities in generating rigidity effects. Applications to the Hilbert transform and related singular integral operators are discussed [3], providing new insights into the interaction between spectral asymmetry and analytic structure.