While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.
New Simpson Type Integral Inequalities for
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-Convex Functions and Their Applications
