Allgemeine Gesichtspunkte zur Interpolation und einige durch sie angeregte Untersuchungen.
In a previous paper I have proved the unicity of the solution of a certain linear partial differential equation of the first order. Here I give sufficient conditions for the existence and calculability of such a solution; moreover, I transform the above equation into a linear integral equation of a new type.
A first order linear partial differential equation is considered in an interval T of the r-dimensional cartesian space. Under suitable hypotheses it is shown that this equation has at most one smooth solution in T. This result generalizes to an arbitrary r a theorem given by the author, since 1930, in the case r = 2.
In considering a pair of ordered real variables (such as appears in fundamental chapters of most treatises of infinitesimal analysis), it is shown that the oscillation interval of one of the two variables is always completely contained in the oscillation interval of the other variable.
A simple unprecedented example is given concerning a linear partial differential equation of an arbitrary order in two real independent variables, a solution of which is determined in a suitably given domain, when its value along an arc lying in it is prescribed.
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