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In a previous paper [4] the following problem was considered:find, in the class of Fourier polynomials of degree n, the one which minimizes the functional:(0.1) $$J^* [F_n ,\sigma ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\frac{{\sigma ^r }}{{r!}}} \left\| {F_n^{(r)} } \right\|^2$$ , where ∥·∥ is theL2 norm,Fn(r) is therth derivative of the Fourier polynomialFn(x), andf(x) is a given function with Fourier coefficientsck. It was proved that the optimal polynomial has coefficientsck* given by(0.2) $$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$ . In this paper we consider the more general functional(0.3) $$\hat J[F_n ,\sigma _r ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\sigma _r \left\| {F_n^{(r)} } \right\|^2 }$$ , which reduces to (0.1) forσr=σr/r!.

The a priori evaluation of the pointwise approximation, by a regularization method suggested earlier, of a $ {L^1}$f, when only noisy Fourier coefficients of f are known, is given here in a formulation more suitable for numerical verification. A careful evaluation of the constants involved in the formulas is also provided. Finally, a procedure is described of a statistical check performed in order to verify whether the theoretically suggested value of the regularization parameter is a good choice with respect to functions generally arising in applications.

Summary: An iterative procedure is given which enables effecting a rather small number of permutations to decide, in a great number of cases, if two matrices can, or cannot, be reduced one to the other by means of permutations on the rows and columns. Some examples of application of this algorithm are given.

Summary: A great number of pseudo-random sequences is studied and results on the statistical behaviour of a selected part of them are given. Some of the considered sequences, studied here for the first time, are aperiodic, uniformly distributed and with good pseudorandomness qualities.