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This paper applies difference operators to conditionally positive definite kernels in order to generate kernel $$B$$-splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel $$B$$-spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel $$B$$-spline is constructed adaptively on the data knot set $$X$$, or we use a fixed difference scheme and shift its associated kernel $$B$$-spline around. In the latter case, the kernel $$B$$-spline so obtained is strictly positive in general. Furthermore, special kernel $$B$$-splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.

In this paper a local type method is proposed to smooth noisy data. Criteria of convergence and error bounds are given.

Recombinant human phenylalanine hydroxylase (hPAH) was produced in high yields in Escherichia coli using the pET and pMAL expression vectors. In the pMAL system, hPAH was fused through the target sequences of the restriction protease factor Xa (IEGR) or enterokinase (D4K) to the C-terminal end of the highly expressed E. coli maltose-binding protein (MBP). The recombinant hPAH, recovered in soluble forms, revealed a high specific activity even in crude extracts and was detected as a homogeneous band by Western-blot analysis using affinity-purified polyclonal rabbit anti-(rat PAH) antibodies. The enzyme expressed in the pET system was subject to limited proteolysis by host cell proteases and was difficult to purify with a satisfactory yield. By contrast, when expressed as a fusion protein in the pMAL system, hPAH was resistant to cleavage by host cell proteases and was conveniently purified by affinity chromatography on an amylose resin. Catalytically active tetramer-dimer (in equilibrium) forms of the fusion protein were separated from inactive, aggregated forms by size-exclusion h.p.l.c. After cleavage by restriction protease, factor Xa or enterokinase, hPAH was separated from uncleaved fusion protein, MBP and restriction proteases by hydroxylapatite or ion-exchange (DEAE) chromatography. The yield of highly purified hPAH was approx. 10 mg/l of culture. The specific activity of the isolated recombinant enzyme was high (i.e. 1440 nmol of tyrosine.min-1.mg-1 with tetrahydrobiopterin as the cofactor) and its catalytic and physicochemical properties are essentially the same as those reported for the enzyme isolated from human liver. The recombinant enzyme, both as a fusion protein and as purified full-length hPAH, was phosphorylated in vitro by the catalytic subunit of cyclic AMP-dependent protein kinase. The phosphorylated from of hPAH electrophoretically displayed an apparently higher molecular mass (approximately 51 kDa) than the non-phosphorylated (approximately 50 kDa) form.

In this paper we study a method for the approximation by a monotone smooth function of a discrete set of observations. By this method it is possible to obtain a completely automatic program. We show that the solution is monotone with probability tending to one and by the test examples that it is sufficient in the practical case.

Summary The problem of the numerical approximation of multivariable functions has been solved by the Monte Carlo method when the data points are assumed to be given on discrete lattice points [5, 8, 2]. When the data points are randomly distributed and very numerous there are some results in the literature [3, 6] but if the number of the points is less than 2k, wherek is the dimension of the space, it is very difficult to develop approximation formulas. This paper gives a solution to this problem by local approximations.

A number of local interpolation methods suitable for statistical and graphical meteorological applications, including weighted averages, least-squares polynomials and successive corrections, were compared in this study. Daily observations of temperature and geopotential height recorded in two years by 71 European stations were used as data sources; the results of the interpolations were checked against the observed values in 19 selected stations, the data of which were assumed as missing. The methods were tested in points with different conditions of data density and isotropy.

Sommario: In questo lavoro viene illustrata una tecnica Monte Carlo per l’interpolazione di funzioni a più variabili nel caso in cui i valori della funzione siano dati su di un reticolo. Tale tecnica sfrutta le funzioni splines cardinali. Sono inoltre riportati alcuni esempi.