A recurrence relation for the square root

The behavior of the sequence x_{n + 1} = x_n(3N − x_n²)/2N is studied for N > 0 and varying real x₀. When 0 < x₀ < (3N)^1/2 the sequence converges quadratically to N^1/2. When x₀ > (5N)^1/2 the sequence oscillates infinitely. There is an increasing sequence β_r, with β₋₁ = (3N)^1/2 which converges to (5N)^1/2 and is such that when β_r < x₀ < β_{r + 1} the sequence {x_n} converges to (−1)^rN^1/2. For x₀ = 0, β₋₁, β₀,… the sequence converges to 0. For x₀ = (5N)^1/2 the sequence oscillates: x_n = (−1)ⁿ(5N)^1/2. The behavior for negative x₀ is obtained by symmetry.     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

A sufficient condition for pairing problem of generators in symmetrizable Kac–Moody algebras

In a symmetrizable Kac–Moody algebra g(A), let α=∑_i=1ⁿk_iα_i be an imaginary root satisfying k_i>0 and α,α_i<0 for i=1,2,…,n. In this paper, it is proved that for any x_αg_α{0}, satisfying [x_α,f_n]≠0 and [x_α,f_i]=0 for i=1,2,…,n−1, there exists a vector y such that the subalgebra generated by x_α and y contains g′(A), the derived subalgebra of g(A).     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

Multi-Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation

This paper studies the Multi-Resolution Analyses of multiplicity d (d *), that is, the families (V_n)_n of closed subspaces in ²( ) such that V_n V_{n + 1}, V_{n + 1} = DV_n, where Dƒ(x) = ƒ(2x), and such that there exists a Riesz basis for V₀ of the form {φ_i(· − k), i = 1, . . . , d,k }, with φ₁, . . . , φ_d V₀. Using the Fourier transform, we prove that (λ) = ^t[ ₁(λ), . . . , _d(λ)] = H(λ/2) (λ/2), where H is in the set _d of continuous 1-periodic functions taking values in (d, ). If d = 1, the definition corresponds to the standard Multi-Resolution Analyses, and one can characterize the regular 1-periodic complex-valued functions H (called, then, scaling filters) which yield a Multi-Resolution Analysis. In this paper, we generalize this study to d ≥ 2 by giving conditions on H _d so that there exists = ^t[ ₁, . . . , _d] in ²( , ^d) solution of (λ) = H(λ/2) (λ/2), and so that the integer translates of φ₁, . . . , φ_d form a Riesz family. Then, the latter span the space V₀ of a Multi-Resolution Analysis of multiplicity d. We show that the conditions on H focus on the zeros of det H(·) and on simple spectral hypotheses for the operator P_H defined on _d by P_HF(λ) = H(λ/2)F(λ/2)H(λ/2)* + H(λ/2 + 1/2)F(λ/2 + 1/2)H(λ/2 + 1/2)*. Finally, we explore connections with the order r dyadic interpolation schemes, where r *.     

Convergence of greedy approximation for the trigonometric system

Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form G_m(f ) := ∑_kЄΛ f^(k) e ^{(i k,x)}, where ΛˆZ^d is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that G_m(f ) gives the best m-term approximant in the L₂-norm and, therefore, for each f ЄL₂, ║f-G_m(f )║₂→0 as m →∞. It is known from previous results that in the case of p ≠2 the condition f ЄL_p does not guarantee the convergence ║f-G_m(f )║_p→0 as m →∞.. We study the following question. What conditions (in addition to f ЄL_p) provide the convergence ║f-G_m(f )║_p→0 as m →∞? In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {A_n}_n=1^∞ to guarantee the L_p-convergence of {G_m(f )} for all f ЄL_p , satisfying a_n (f ) ≤A_n , where {a_n (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄL_p and ║G_M(m)(f ) - G_m(f )║_p →0 as m →∞ imply ║f - G_m(f )║_p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞.     

Abschätzungen der unvollständigen Beta-Funktion in mehreren reellen Variablen und Parametern

The integral ofu ₁ ^p–1 ...u _n ^pn–1 (1–u₁–...–u_n)^q, extended over then-simplex {(u ₁,...,u _n) ₊ ⁿ :u ₁/x ₁+...+u _n/x _n<1}, is estimated from below under the condition thatp ₁+...+p_n+qc=p₁/x₁+...+p_n/x_n=const.     

Hardy-Littlewood inequalities for Ciesielski-Fourier series

Summary The inequality (^∞_n=2Σn^p-2│^{^}f(n) │^p)≤C_p││f││_H (0 is proved for the one- and two-dimensional Ciesielski-Fourier coefficients of functions in Hardy spaces.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.     

On the Behaviour of Zeros of Jacobi Polynomials

Denote by x_n,k(α,β) and x_n,k(λ)=x_n,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P^(α,β)_n(x) and of the ultraspherical (Gegenbauer) polynomial C^λ_n(x), respectively. The monotonicity of x_n,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P^(a,b)_n(x) are smaller (greater) than the zeros of P^(α,β)_n(x) are provided. A. Markov proved that x_n,k(a,b)<x_n,k(α,β) (x_n,k(a,b)>x_n,k(α,β)) for every n and each k, 1kn if a>α and b<β (a<α and b>β). We prove the converse statement of Markov's theorem. The question of how large the function f_n(λ) could be such that the products f_n(λ)x_n,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that f_n(λ)=(λ+(2n²+1)/(4n+2))^1/2 obeys this property. We establish the sharpness of their result.     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

Companions of the inequalities of Fejér--Jackson and Young

Summary Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities -1<ⁿ_k-1Σ(cos(kx)+sin(kx))/k (n ≥1; 1<x<π) and -1/2Si(π)<ⁿ_k-1Σ(cos(kx)·sin(kx))/k (n ≥1; xЄR) We prove that the following counterpart is valid for all integers n ≥1 and real numbers xЄ (0, π): -3/2≤ⁿ_k-1Σ(cos(kx)-sin(kx))/k where the sign of equality holds if and only if n =2 and x = π /2.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Primitive polynomial with three coefficients prescribed

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a₁,a₂,a₃)F_q³, there exists a primitive polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿσ_n over F_q of degree n with the first three coefficients σ₁,σ₂,σ₃ prescribed as a₁,a₂,a₃ when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over F_q with the first three coefficient prescribed where the characteristic of F_q is 2 or 3.