Direct form seminorms arising in the theory of interpolation by translates of a basis function (Q1966316)

The authors study here seminorms associated with some interpolation problems, particularly those involving radial basis functions. It is observed that in many applications of the theory, the seminorms arise in the Fourier transform setting [cf. \textit{W. A. Light} and \textit{H. S. J. Wayne}, Numer. Math. 81, 415-450 (1999; Zbl 0918.65007)]. The authors show that most of the foregoing seminorms can be rewritten directly in terms of the function (rather than their Fourier transforms). For advantages in writing the seminorms in terms of the functions one could refer to \textit{J. Duchon} [RAIRO, Anal. Numer. 12, No. 4, 325-334 (1978; Zbl 0403.41003)].