Hilbert’s Inequality and Compensating Difficulties Một số kinh nghiệm đáp ứng hầu hết trong vấn đề giải quyết diễn ra khi một người bắt đầu trên một con đường tự nhiên và sau đó va chạm vào một khó khăn bất ngờ. Nhân dịp này nhìn sâu sắc hơn về vấn đề này buộc chúng ta phải tìm một cách tiếp cận hoàn toàn mới. Có lẽ thường chúng ta chỉ cần tìm một cách để báo chí khó khăn hơn trên một biến thể thích hợp của kế hoạch ban đầu. Vấn đề giới thiệu của chương này cung. | 10 Hilbert s Inequality and Compensating Difficulties Some of the most satisfying experiences in problem solving take place when one starts out on a natural path and then bumps into an unexpected difficulty. On occasion this deeper view of the problem forces us to look for an entirely new approach. Perhaps more often we only need to find a way to press harder on an appropriate variation of the original plan. This chapter s introductory problem provides an instructive case here we will discover two difficulties. Nevertheless we manage to achieve our goal by pitting one difficulty against the other. Problem Hilbert s Inequality Show that there is a constant C such that for every pair of sequences of real numbers an and bn one has ambn y m n V m 1 n 1 xm 1 2 2 m Zbn . n 1 Some Historical Background This famous inequality was discovered in the early 1900s by David Hilbert specifically Hilbert proved that the inequality holds with C 2n. Several years after Hilbert s discovery Issai Schur provided a new proof which showed Hilbert s inequality actually holds with C n. We will see shortly that no smaller value of C will suffice. Despite the similarities between Hilbert s inequality and Cauchy s inequality Hilbert s original proof did not call on Cauchy s inequality he took an entirely different approach that exploited the evaluation of some cleverly chosen trigonometric integrals. Nevertheless one can prove 155 156 Hilbert s Inequality and Compensating Difficulties Hilbert s inequality through an appropriate application of Cauchy s inequality. The proof turns out to be both simple and instructive. If S is any countable set and as and 3s are collections of real numbers indexed by S then Cauchy s inequality can be written as 1 1 Y ps fe a2p T 2 . sES VsES 2 vs S 7 This modest reformulation of Cauchy s inequality sometimes helps us see the possibilities more clearly and here of course one hopes that wise choices for S as and 3s will lead us from the bound
