Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

Let W be a finite-dimensional Z/p-module over a field, k, ofcharacteristic p. The maximum degree of an indecomposable elementof the algebra of invariants, k[W]^Z/p, is called the Noethernumber of the representation, and is denoted by rß(W).A lower bound for rß(W) is derived, and it is shownthat if U is a Z/p submodule of W, then rß(U) rß(W).Aset of generators, in fact a SAGBI basis, is constructed fork[V₂ V₃]^Z/p, where V_n is the indecomposable Z/p-module of dimensionn. 2000 Mathematics Subject Classification 13A50, 20J06.     

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

The Adamjan-Arov-Krein Theorem on Hp-Spaces (2 <= p <= {infty}) and on the Disc Algebra

This paper is devoted to the following extension of the AAKtheorem. Let (p,q)[2, + ]². Let u: H^p x H^q C bea hankelianbilinear form and n N*. There is a hankelian bilinear formv: H^p x H^q C with rk(v) < n and ||u–v|| Ca_n(U) forsome constant C > 0 depending only on (p,q). Moreover, H^por H^q may be replaced by A in this statement.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

Classification of Quantum Groups SUq(n)

It is shown that the compact matrix quantum groups SU_q(2) arenon-isomorphic to each other for q[–1, 1]\{0}, and thatthe compact matrix quantum groups SU_q(n) are non-isomorphicto each other for q(0, 1]. Some invariants for compact quantumgroups are also discussed.