Hilbert s Inequalities

Transcription

1 Hilbert s Inequalities 1. Introduction. In 1971, as an off shoot of my research on the Davenport Halberstam inequality involving well spaced numbers, I took the numbers to be equally spaced and was led to the inequality cosec π(r s) ( 1) x r 2 (1) for all complex numbers x 1,..., x. If we let n N and n and take x r = 0 for n < r, then dividing both sides of inequality (1) by and letting gives Hilbert s second inequality n x r 2. (2) r s π n (See Hardy, Littlewood, Pólya [1].) On reading my manuscript, Hugh Montgomery observed that a strengthening of Hilbert s inequality could be obtained: r s π(1 1 ) x r 2. (3) Montgomery conjectured that if the largest eigenvalue of the Hilbert matrix is iµ, µ > 0, then π µ c log. He was able to obtain a weaker order of magnitude result. I have not seen a reference to (3). We remark that the first Hilbert inequality r,s=1 r + s π x r 2. (4) 1

2 is much less mysterious and also follows with some care, from inequality (1). Also (see Wilf [4, pages 2 5]) if λ is the largest eigenvalue of the corresponding (positive definite) Hilbert matrix, then Theorem 2.2 of Wilf [4] states that λ = π π 5 /2(log ) 2 + O(log log /(log ) 3 ). 2. A skew circulant matrix. LEMMA 1. The eigenvalues of the skew circulant matrix complex matrix a 0 a 1 a 1 a 1 a 0 a 2 A =.... a 1 a 2 a 0 are given by 1 λ s = a r e (2s 1)rπi, s = 1,...,. r=0 Proof. (Montgomery) egard a 0,..., a 1 as indeterminates. Let X s, s = 1,..., be the column vector with entries Then direct calculation reveals that e (2s 1)rπi, r = 0,..., 1. AX s = ( 1 r=0 a r e (2s 1)rπi ) X s. LEMMA 2. Let C = [c rs ] be the matrix defined by { cosec π(r s) if c rs = 0 if r = s. Then the eigenvalues of C are the purely imaginary numbers (2s 1 )i, s = 1,...,, 2

3 or in other words, the numbers ±( 1)i, ±( 3)i,... POOF. In the nomenclature of Lemma 1, C = A, where a 0 = 0 and a r = cosec πr, r = 1,..., 1. Hence the eigenvalues of C are given by λ s = 1 = i e (2s 1)rπi cosec πr sin (2s 1)πr sin πr From the first of these expressions for λ s we deduce that λ ν+1 λ ν = 2i. Also λ 1 = ( 1)i. Consequently the theorem follows. Noting that the eigenvalue of largest modulus of C is i( 1), we have the following result: COOLLAY. For all complex numbers x 1,..., x, cosec r s π(r s). ( 1) x r 2. POOF. A skew symmetric matrix C is a normal matrix and is hence unitarily similar to a diagonal matrix. We then argue as in the proof of Theorem , Mirsky [2, page 388]. 3. An improvement to Hilbert s inequality. The next result is due to Schur (see Satz 5, Mirsky [3, page 11].) LEMMA 3. Let C = [c rs ] and D = [d rs ] be matrices with D positive definite Hermitian. Then if µ = max d rr and ν is a positive number such r that the inequality c rs ν x r 2 s=1 holds for all complex numbers x 1,..., x, then the inequality s=1 c rs d rs µν 3 x r 2

4 holds for all complex numbers x 1,..., x. We are now able to derive the improvement in Hilbert s second inequality, as pointed out by Montgomery: THEOEM. The inequality r s π(1 1 ) x r 2 (5) for all complex numbers x 1,..., x. POOF. We have r s = s=1 c rs d rs, (6) where c rs = d rs = { cosec π(r s) if 0 if r = s. { 1 r s π(r s) sin π if if r = s. It is easy to prove that D is positive definite. For π d rs = d rs + π r s r s r 1 2 = = r r s x r 2 e 2π(s r)ix dx x r e 2πrix 2 dx. Since d rr = π/, we have µ = π/. Also by the Corollary, we may take ν = 1. Consequently (5) follows from (6) and Lemma 3. 4

5 Dr. Graham Jameson has supplied another proof of the Theorem, which does not use Lemma 3. Let and let b r,s = { 1/(r s) if, 0 if r = s, S = s=1 b r,s x r x s. With the notation of the paper, b r,s = c r,s d r,s. Write (as usual) e(t) = e 2πit. Note first that 1/2 1/2 e(λt) dt = sin πλ πλ for λ 0. Hence d r,s = π = π 1/2 1/2 1/2 1/2 ( ) (r s)t e g r (t)g s (t) dt, dt where g r (t) = e(rt/) (also when r = s). So S = π 1/2 1/2 s=1 c r,s g r (t)g s (t) x r x s dt. By (1), applied to the scalars g r (t)x r, we have c r,s g r (t)x r g s (t)x s ( 1) x 2. s=1 Hence S π ( ( 1) x 2 = π 1 1 ) x 2. 5

6 eferences [1] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press [2] L. Mirsky, An introduction to linear algebra, Oxford University Press [3] I. Schur, Bemerkungungen zur Theorie der beschränkten Bilinearformen mit endlich vielen Verändlichen, J. reine angew. Math., 140 (1911), [4] H.S. Wilf, Finite sections of some classical inequalities, Ergebnisse der Mathematik, Band 52, Springer