G. J. O. Jameson

Positive and minimal projections in function spaces

If L is a linear lattice and E is a linear subspace of L, it is natural to ask whether there is a positive projection of L onto E (a projection P is positive, or monotone, if x > 0 implies Px > 0). This is always the case, for example, when L is Lp(u) and E is a closed linear sublattice [4, Chap. 31. However, much less is known about the situation when L is the function space C(X) (X compact, Hausdorff) with supremum norm, though for certain subspaces Korovkin’s theorem implies that there is no positive projection. In Section 2, we give necessary and sufftcient conditions for there to be a positive projection of a normed linear lattice L onto an n-dimensional subspace L,. As a corollary, we see that if M is a closed sublattice of a Banach lattice L and there is a positive projection of M onto L,, then there is a positive projection of L onto L,. In particular, every finite-dimensional sublattice of L admits a positive projection. When L is C(X), our characterization reduces to the following: L, admits a positive projection if and only if there exist positive functions b, ,..., b, in L, and points x, ,..., x, of X such that b,(x,) = 6,. In Section 3, we study the companion problem for finite-codimensional subspaces of C(X). We prove, in fact, that if X has no isolated points, then such subspaces never admit positive projections.

Inequalities for Gamma Function Ratios

Abstract A short proof is given of a comprehensive system of inequalities for including results obtained independently by Artin, Wendel, and Gautschi. Applications include inequalities for binomial coefficients and the Bohr–Mollerup theorem.

Inequalities for the perimeter of an ellipse

The perimeter of the ellipse x 2 / a 2 + y 2 / b 2 = 1 is 4 J (a, b) , where J (a, b) is the ‘elliptic integral’ This integral is interesting in its own right, quite apart from its application to the ellipse. It is often considered together with the companion integral Of course, we may as well assume that a and b are non-negative.

Series involving ζ(n)

Recall that for integers n ≥ 2, ζ(n) is defined by ζ(n) = ∞ k=1 1 k n = 1 + 1 2 n + 1 3 n + · · ·. Of course, ζ(1) is not defined, since ∞ k=1 1 k is divergent. A well-known particular value is ζ(2) = π 2 /6: numerous alternative proofs of this fact have been presented in the Gazette, e.g. the recent notes [1], [2]. Here we will consider infinite series of the form ∞ n=2 a n ζ(n) or ∞ n=2 a n [ζ(n) − 1]. There are cases where the second type is convergent, while the first one is divergent. In fact, this is already seen in the case a n = 1: clearly, ζ(n) > 1 for all n, so ∞ n=2 ζ(n) is divergent. However, ζ(n) − 1 = 1 2 n + 1 3 n + · · · , and from this it is quite easy to show (for example by integral estimation) that ζ(n)−1 < 1 2 n−1 for all n ≥ 3, so that ∞ n=2 [ζ(n) − 1] is convergent. We will not bother with the details here, because this fact will emerge with no extra effort from the reasoning below. Our investigation will take us on a round trip of several of the most basic infinite series (consequently, this topic can serve as a rather good revision course on series for students). By definition, ∞ n=2 a n ζ(n) = ∞ n=2 ∞ k=1 a n k n. To make further progress, the next step is inevitably to reverse the order of summation in this double series. Since everything will depend on it, we pause here to consider the validity of this reversal. In general terms, consider the repeated series ∞ n=1 ∞ k=1 a n,k .

The interpolation proof of Grothendieck's inequality

This note is an exposition of the simple and elegant approach to Grothendiecks inequality given in [ 2 ] and lsqb; 4 ], with one further simplification. The process of factorizing through L 2 ([ 2 ], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.

2-convexity and 2-concavity in Schatten ideals.

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, E) is 2-convex, then there is another Banach sequence space (F, F) such that x; = x2F for all x e E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

Some inequalities for (a + b)p and (a + b)p + (a − b)p

We start from two simple identities: For any p > 0 and 0 ≥ b ≥ a , now let Can we formulate statements about G p ( a, b ) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form A p a p ≤ G p ( a, b ) ≤ B p a p for certain constants A p and B p , and for (2) a similar statement with a p replaced by a p + b p . However, these are not the only possibilities, as we shall see.

94.03 Two squares and four squares: the simplest proof of all?

Numerous proofs are now known for both theorems. Three of the best known methods for the two-squares theorem, all completely different, can be seen in [3], chapter 10: they use, respectively, (a) the “method of descent”, (b) the Gaussian integers, (c) the geometry of lattices in the plane. Methods similar to (a) and (c) are then described for the four-squares theorem: (a) is essentially Euler’s method, and it is the one found in most textbooks.

Inequalities comparing

Consider the comparison between (a + b)p and a p + bp, where a, b and p are positive. It is elementary that (a + b)p > a p + bp for p > 1 and the opposite holds for 0 < p < 1 (let b/a = x ≤ 1: then for p > 1, we have (1 + x)p > 1 + x > 1 + x p). Let us write Fp(a, b) = (a + b)p − a p − bp. For p = 2, 3, we have the identities F2(a, b) = 2ab, F3(a, b) = 3(a2b + ab2). Also, when b/a is small, (a + b)p is approximated by a p + pa p−1b. These facts suggest that it is a natural idea to look for estimates of Fp(a, b) in terms of G p(a, b) = a p−1b + abp−1.

(a+b)^p−a^p−b^p

The identity in question really is simple: it says, for u ≠ −1, We describe two types of definite integral that look quite formidable, but dissolve into a much simpler form by an application of (1) in a way that seems almost magical. Both types, or at least special cases of them, have been mathematical folklore for a long time. For example, case (10) below appears in [1, p. 262], published in 1922 (we are grateful to Donald Kershaw for showing us this example). However, they do not seem to figure in most books on calculus except possibly tucked away as an exercise The comprehensive survey [2] mentions the second type on p. 253, but only as a lemma on the way to an identity the authors call the ‘master formula’ We come back to this formula later, but only after describing a number of other more immediate applications.