Notes on linear transformations. I
Einar Hille · Transactions of the American Mathematical Society · 1936
Under the above title the author intends to publish some investigations on the properties of linear transformations in abstract spaces.In the present note the space is a suitable subset of the set of all measurable functions defined for -oo <x< oo, and the transformations are of the form(1)The results, which are somewhat loosely knit together, cluster around four problems, (i) The originators of zero, i.e., the solutions of the equation ( 2)(ii) The invariant elements, i.e., the solutions of the equation(3) Ka[f\=f.(iii) The functional equations satisfied by Ka [/] for special choices of the kernel, (iv) The metric properties of the transformation Ka[f], including properties of contraction, and degree of approximation of/by A"[/] for large values of a.The material is grouped as follows.§1 gives a survey of problems (i), (ii) and (iv) for a general kernel A(w)eLi(-°°, o°), A(«) SjO.It lies in the nature of things that the results for this case are rather incomplete.They probably do not offer much of any novelty to the workers in the field, but serve as background for the discussion in § §3-4.The existence of functional equations obtained by superposition is established in §2, and the equations are given for four particular kernels which may be associated with the names of Dirichlet, Picard, Poisson, and Weierstrass.A closer study of the last two kernels, which satisfy the same functional equation, is given in §3, whereas the kernel of Picard is treated in §4.It turns out that the study of problems (i), (ii)and (iv) for these special kernels is much simplified by the corresponding functional equations.Some results on the Dirichlet kernel occur in §5, but lack the same degree of completeness, sharpness and simplicity, f