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This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the aforementioned context), concepts of qualifications as well as order optimal rates of convergence are presented. Optimality results under general source conditions expressed in terms of index functions are also studied. Finally, numerical experiments on three testarXiv:1909.12020v2 fatcat:msb3ap3z5nbxzo7kac5fsjqe44