Summary: We consider the recovery of the impedance A(y) for a one-dimensional seismic problem \(v_{tt}=(Av_ y)_ y/A\), where y is the travel time, with impulse input \(v(y,0)=\delta (y)\), and readout \(v(0,t)=\delta (t)+g(t)\), \(g(t)=\) \((2/\pi)\int^{\infty}_{0}\hat g(\lambda)Cos \lambda t d\lambda,\) \(\hat g\) being the Fourier cosine transform. First, it follows that, for \(\Delta\) denoting change, \(| \Delta A(y)| \leq \tilde c(y)\| \Delta g\|_{\infty}\) on [0,2y]. The problem of estimating \(\| \Delta g\|_{\infty}\) on [0,2y] is ill-posed, if only estimates \(\| \Delta g\|_ 2\leq \epsilon\) can be obtained. Using regularization and summability, we construct a sequence of approximants \(g^*_ n\) which converges to g pointwise on [0,2y] uniformly. Thus, \(g^*_ n\to g\) in \(L^{\infty}(0,2y)\) and the corresponding \(A_ n(y)\) converse to A(y) pointwise.
This paper also serves as an introduction to the connection of regularization and summability ideas with transmutation formulas. Applications to approximation theory for generalized eigenfunction expansions for special functions can also be envisioned.