This paper investigates an ill-posed boundary value problem for the wave equation with continuous spectrum, where the solution fails to depend continuously on the boundary data. We propose a novel spectral filter regularization method to stabilize the reconstruction. Two filter functions are introduced, designed to suppress unstable high-frequency components in the frequency domain. Under appropriate a priori parameter choice rules, we establish convergence estimates of H¨ older type. Specifically, we prove that the regularized solution converges to the exact solution uniformly in time with explicit error bounds. We also demonstrate the continuous dependence of the regularized solution on the data and provide convergence rates under suitable regularity assumptions on the initial data. The proposed method extends classical spectral regularization techniques by introducing flexible filter parameters that allow improved control over the trade-off between stability and accuracy. Unlike existing approaches, our filters are constructed to handle the singularities arising from the zeros of the sine kernel in the continuous spectrum setting. A detailed comparison with the work of Gal and Gal (2008) is provided, showing that our method offers explicit regularization, quantitative convergence rates, and uniform convergence in time, whereas their approach only establishes well-posedness under restrictive conditions and does not provide regularization strategies when these conditions fail.

This paper investigates an ill-posed boundary value problem for the wave equation with continuous spectrum, where the solution fails to depend continuously on the boundary data. We propose a novel spectral filter regularization method to stabilize the reconstruction. Two filter functions are introduced, designed to suppress unstable high-frequency components in the frequency domain. Under appropriate a priori parameter choice rules, we establish convergence estimates of H¨ older type. Specifically, we prove that the regularized solution converges to the exact solution uniformly in time with explicit error bounds. We also demonstrate the continuous dependence of the regularized solution on the data and provide convergence rates under suitable regularity assumptions on the initial data. The proposed method extends classical spectral regularization techniques by introducing flexible filter parameters that allow improved control over the trade-off between stability and accuracy. Unlike existing approaches, our filters are constructed to handle the singularities arising from the zeros of the sine kernel in the continuous spectrum setting. A detailed comparison with the work of Gal and Gal (2008) is provided, showing that our method offers explicit regularization, quantitative convergence rates, and uniform convergence in time, whereas their approach only establishes well-posedness under restrictive conditions and does not provide regularization strategies when these conditions fail.