A General Moment Problem

Boas, R. P.

doi:10.2307/2371530

Cited by 60 publications

(35 citation statements)

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“…By changing variables we see that this is equivalent to the inequality If b n = 1 for M + 1 < n < M + N, b n = 0 otherwise then we arrive again at (6). If b n > 1 for M + 1 < n < M + AT then (7) gives (2) were the first to consider the large sieve via duality.…”

Section: (7)mentioning

confidence: 87%

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The analytic principle of the large sieve

Montgomery¹

1978

Bull. Amer. Math. Soc.

165

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E. Bombieri [12] has written at length concerning applications of the large sieve to number theory. Our intent here is to complement his exposition by devoting our attention to the analytic principle of the large sieve; we describe only briefly how applications to number theory are made. The large sieve was studied intensively during the decade [1965][1966][1967][1968][1969][1970][1971][1972][1973][1974][1975], with the result that the subject has lost its mystery: We now possess a variety of simple ideas which provide very precise results and a host of variants. While the large sieve can no longer be considered deep, it nevertheless gives powerful estimates in many different settings. ) has obtained k * 2. In all of these arguments the large sieve is a major tool.) The large sieve remained the province of a few specialists, until the appearance in 1965 of a fundamental paper of K. F. Roth [86], followed immediately by a major contribution of Bombieri [7]. As we consider it here, the large sieve was first reduced to its basic analytic principle by H. Davenport and H. Halberstam [21].2. The nature of the large sieve. For M + 1 < n < M + AT we let a n be arbitrary complex numbers, and we form the trigonometric polynomial M+N S (a) -2 a n e(na);here e(9) = e 2 ™ e , so that S (a) has period 1. Let a" ..., a R be points which are well spaced (mod 1) in the sense that This is an expanded account of an invited address entitled The large sieve for the mathematician in the street,

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Section: (7)mentioning

confidence: 87%

“…= p 2 = 2 of Lemma 2. In order to have an inequality of this sort it is not necessary to assume that the q> r are orthonormal; we have LEMMA 3 (BOAS [6] Note that if the <p r are orthonormal then we clearly have B = 1 in (v), so that (iv) gives Bessel's inequality.…”

Section: (7)mentioning

confidence: 99%

The analytic principle of the large sieve

Montgomery¹

1978

Bull. Amer. Math. Soc.

165

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“…The proof is based on the theory of compact operators in Hilbert space together with the following elegant characterization of moment sequences due to Boas [3].…”

Section: On a Class Of Riesz-fischer Sequencesmentioning

confidence: 99%

On a class of Riesz-Fischer sequences

Young

1998

Proc. Amer. Math. Soc.

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“…Theorem 1. Let {v¡}, i =1,2,"i,..., be a sequence of elements of a Hilbert space H, and suppose that the inequality (1) d2"£a2 <\\£aiVi holds for every finite equation ofscalars {a¡} . If an element v0 is adjoined to {Vj}, then the resulting set satisfies, for every finite sequence ofscalars, (…”

Section: An Estimate Of the Norm Of A Linear Combinationmentioning

confidence: 99%

A Pythagorean inequality

Reid¹

1995

Proc. Amer. Math. Soc.

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Abstract. Let {vx, v2 , v$ , ... } be a sequence of elements of a Hilbert space, and suppose that (one or both of) the inequalities d2J2aj < \[52a¡v¡\\2 < D2 £] aj hold for every finite sequence of scalars {a,} . If an element v0 is adjoined to {v¡} , then the resulting set satisfies (one or both of) d2JA,af < Eai'vill2 ^ ^o ¿A,a2 » where, denoting the norm of vo by r and its distance from the closed linear span of the v¡ by ô , d2 = d2 and + Ur2-d2-yV2 + d2)2 -4d262\ D2 = D2 + X-(r2 -D2 + y/(r2 + D2)2 -^D2S2\ .Both bounds are best possible. If Vq is in the span of the original set, the expressions above simplify to do = 0 and D2, = D2 + r2 . If the original set is a single unit vector vx , so d = D = 1 , and if ^o-Lvi is a unit vector so ô = X , then the above is (a2 + b2) < \\avo + bvx\\2 < (a2 + b2), the Pythagorean Theorem.Several consequences are deduced. If v¡ are unit vectors, 5Z aJ = 1 , and (5, is the distance from v¡ to the span of its predecessors (so that the volume of the parallelotope spanned by the v¡ is V" = 6xô2 ■ ■ ■ ôn), the above result is used to show that ||¿"=oa>v

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A General Moment Problem

Cited by 60 publications

References 0 publications

The analytic principle of the large sieve

The analytic principle of the large sieve

On a class of Riesz-Fischer sequences

A Pythagorean inequality

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