Theorem of Lindeberg in the calculus of variations
Transactions of the American Mathematical Society
1. Introduction. The present paper is the second of a sequence of three papers concerned with the isoperimetric problem of Bolza in parametric form. The first paper(') is concerned with certain properties of the Weierstrass ¿-function that will be useful in the present paper and in the one to follow. The third paper will be concerned with a sufficiency theorem conjectured by McShane(2). In the present note is found an extension of the theorem of Lindeberg and related results. This theorem is an
... obvious extension to the parametric case of a similar theorem given by Reid (3)(4) for the nonparametric case and is an immediate consequence of the arguments used by Reid in an expansion proof for the parametric problems which has not been published as yet. The approach to this theorem here given is different from that given by Reid. Moreover the theorem is stated so as to bring out a condition of uniformity that simplifies the applications of the theorem. Among the many consequences of the theorem of Lindeberg and a related theorem is the equivalence of the sufficiency theorems for the problems of Mayer and Bolza, a result that does not appear to have been established completely heretofore. It is shown moreover that the sufficiency theorem for the isoperimetric problem of Bolza can be obtained from those for the problem of Bolza without isoperimetric conditions. It is also pointed out that the sufficiency theorems for parametric problems can be obtained from those for the nonparametric problems. In addition to further results of similar nature, we give in the last section a result that will be useful in the paper to follow.