tailieunhanh - THE CAUCHY – SCHWARZ MASTER CLASS - PART 11

Hardy’s Inequality and the Flop Flop là một thao tác đại số đơn giản, nhưng nhiều người làm chủ nó cảm thấy rằng họ là mãi mãi thay đổi. Điều này không phải là để nói rằng thất bại là đặc biệt kỳ diệu, trong thực tế, nó là hoàn toàn bình thường. Những gì có thể phân biệt các flop trong số các kỹ thuật toán học là nó hoạt động ở hai cấp độ: đó là chiến thuật ở chỗ nó chỉ là một bước trong một cuộc tranh luận, và đó là chiến lược ở chỗ. | 11 Hardy s Inequality and the Flop The flop is a simple algebraic manipulation but many who master it feel that they are forever changed. This is not to say that the flop is particularly miraculous in fact it is perfectly ordinary. What may distinguish the flop among mathematical techniques is that it works at two levels it is tactical in that it is just a step in an argument and it is strategic in that it suggests general plans which can have a variety of twists and turns. To illustrate the flop we call on a concrete challenge problem of independent interest. This time the immediate challenge is to prove an inequality of . Hardy which he discovered while looking for a new proof of the famous inequality of Hilbert that anchored the preceding chapter. Hardy s inequality is now widely used in both pure and applied mathematics and many would consider it to be equal in importance to Hilbert s inequality. Problem Hardy s Inequality Show that every integrable function f 0 T R satisfies the inequality i i f u du dx 4 f2 x dx J0 Ix J 0 J J0 and show moreover that the constant 4 cannot be replaced with any smaller value. To familiarize this inequality one should note that it provides a concrete interpretation of the general idea that the average of a function typically behaves as well or at least not much worse than the function itself. Here we see that the square integral of the average is never more than four times the square integral of the original. 166 Hardy s Inequality and the Flop 167 To deepen our understanding of the bound we might also see if we can confirm that the constant 4 is actually the best one can do. One natural idea is to try the stress testing method page 159 which helped us before. Here the test function that seems to occur first to almost everyone is simply the power map x xa. When we substitute this function into an inequality of the form i i f u du dx C i f 2 x dx J0 Ix J 0 J J0 we see that it implies 1C - 1 2 2 _ _ 1 2 _ _ 1