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Natural outputs and global inputs of linear systems with a finite number of input data

We shall introduce a new concept for the treatment of certain general linear systems: for a finite number of input data, we would like to consider some natural global output and the corresponding global natural input satisfying the input and output relation. We will be able to formulate such a problem by considering the generalized inverse by means of the Tikhonov regularization in the framework of reproducing kernel Hilbert spaces. We shall give concrete examples for the heat conduction problem and the wave equation case.