Einar Hille

Non-oscillation theorems

where F(x) is a real-valued function defined for x> 0 and belonging to L(E, 1/E) for each E>0. A solution of (1.1) is a real-valued function y(x), absolutely continuous together with its first derivative, which satisfies the equation for almost all x, in particular at all points of continuity of F(x). We shall say that the equation is non-oscillatory in (a, oo), a>O, if no solution can change its sign more than once in the interval. Since the zeros of linearly independent solutions separate each other, it is sufficient that there exists a solution without zeros in the interval in order that the equation be non-oscillatory there. It is well known that (1.1) is non-oscillatory in (0, oo) if F(x) _ 0, but it may also have this property when F(x) ?0 as is shown by the example

On the characteristic values of linear integral equations

on the basis of the general analytic properties of the kernel K (x, ~) such as im tegrability, continuity, differentiability, analyticity and the like? The l i tera ture where this and analogous questions are t rea ted is v e r y considerable [HELLI~G~n-To~eLITZ, I]. ~ A relatively small par t of this l i terature, however, has points of contact with the present paper, the discussion of the majori ty of papers published on the subject being based on various special properties of the kernels. I t is assumed frequent ly tha t the kernel belongs to some special class of functions, or tha t it coincides with the Green s funct ion of a different ial or integro-differential boundary value problem. Problems of this sort will be excluded f rom the scope of our paper a l though they are in teres t ing f rom a theoret ical point of view and impor tan t for the applications.

Remarks on a paper by Zeev Nehari

then f {z) is univalent f or \z\ < 1 . The object of the present note is to show that 2 is the best possible constant in (2) in the following sense: For every C>2 there exists a function f (z) such that f or \z\ < 1 we have (i) f(z) is holoniorphic, (ii) f{z) takes on the value one infinitely often, and (iii) | {f(z), z) | ^ C[\ — | z\ 2 ] ~ 2 with equality f or real values of z. An explicit example of such a function is given by

Pathology of infinite systems of linear first order differential equations with constant coefficients

SummaryThe properties of solutions of finite systems are analyzed and it is shown by examples how these properties may be distorted and ultimately lost in passing from finite to infinite systems.

Remarks On A Known Example of A Monotone Continuous Function

(1929). Remarks On A Known Example of A Monotone Continuous Function. The American Mathematical Monthly: Vol. 36, No. 5, pp. 255-264.

Notes on Linear Transformations. II. Analyticity of Semi-Groups

(1.1.4) T4[To x] = To+x a, f > 0. Here Wax is the Gauss-Weierstrass singular integral and Pax is the Poisson integral for the upper half-plane. Poissons integral for the circle leads to the same functional equation provided we replace the customary parameter r by ea. The equation is also encountered in the applications of Abels method of summation or its various generalizations (convergence factors e-aXn). As an example of a different type, leading to the same equation, we mention the theory of fractional integration.

Notes on linear transformations. I

(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

Contributions to the theory of Hermitian series. II. The representation problem

L Introduction. In the first note of this series [2]f the author laid a broad foundation for the theory of Hermitian series in the complex domain. A number of questions encountered during this investigation were merely mentioned and detailed discussion had to be postponed till later communications. The representation or expansion problem was such a question, that is, the problem of finding necessary and sufficient conditions in order that an analytic function shall be representable by an Hermitian series for complex values of the variable. This problem is solved in the present note. Let ff»(z) denote the wth normalized orthogonal function of Hermite

Oscillation theorems in the complex domain

The aim of the present paper is to throw some light on the question of the distribution in the complex domain of the zeros of functions satisfying linear homogeneous differential equations of the second order. The real zeros of such functions are well known from the works of numerous mathematicians from Sturm and Liouville down to living writers, but our knowledge of the complex zeros is very deficient. It is only in special cases that progress has been made. The number of complex zeros of a hypergeometric function in the case of real parameters has been determined by Hurwitz, Van Vleck and Schafheitlin. Hurwitz has also investigated Bessel functions. The same field has been covered by Macdonald, Porter and Schafheitlin. Hurwitz used in his paper on Bessel functions+ certain integral equalities analogous to those frequently used for establishing the reality of the characteristic values in a boundary problem. ? In his thesis II the present writer used similar equalities for the study of the zeros of Legendre functions. The present paper contains a systematic study of integral equalities, called Greens transforms, which are adjoined to linear differential equations of the second order. It is shown that these equalities give information concerning the distribution of the zeros of a function satisfying such an equation. From the knowledge that a particular solution of the equation in question vanishes at a point in the complex plane, is real on an interval, or similar information, we are able to assign certain regions of the plane, containing the point or the interval where this particular solution cannot vanish. In such a fashion we can estab-

Some remarks on Briot-Bouquet differential equations, II

For the coefficient of the (n k)th power of the derivative we write simplyp,(w). It is a polynomial in w of degree Sk where 6, is <2/z in the first order case and <312 in the second order one. These normal forms are necessary but not sufficient if the equations are not allowed to have solutions with movable branch-points. The present note is devoted to a further study of the solutions using the diagram of Puiseux and analytic continuation. In particular, the validity of Painleve’s determinateness theorem is considered. See [5, pp. 87-891.