Depth of functions of the <Emphasis Type="Italic">k</Emphasis>-valued logic in infinite bases

The realization of functions of the k-valued logic by circuits is considered over an arbitrary infinite complete basis B. The Shannon function D _B (n) of the circuit depth over B is examined (for any positive integer n the value D _B (n) is the minimal depth sufficient to realize every function of the k-valued logic of n variables by a circuit over B). It is shown that for each fixed k ≥ 2 and for any infinite complete basis B either there exists a constant α ≥ 1 such that D _B (n) = α for all sufficiently large n, or there exist constants β (β > 0), γ, δ such that βlog₂ n ≤ D _B (n) ≤ γlog₂ n + δ for all n.     

Composition Operators on the Bloch Space of Several Complex Variables

Abstract In this paper, we study the boundedness and compactness of composition operator C _φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B _n, we give the necessary and sufficient conditions for a composition operator C _φ to be compact on β(B _n) or β ₀(B _n). Supported by the National Natural Science Foundation and the National Education Committee Doctoral Foundation     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Sulle equazioni differenziali lineari soddisfatte dal prodotto di integrali particolari di due equazioni differenziali lineari omogenee assegnate e su alcune formule integrali dei polinomi di Laguerre e di Hermite

Sunto Stabilita l'equazione soddisfatta dal prodotto degli integrali particolari di due generali equazioni differenziali lineari omogenee del 2° ordine assegnate, con procedimento che ha carattere generale, si deducono poi particolari equazioni soddisfatte da cL _n ^(α )(x)L _m ^(β )(x), cH_n(x)H_m(x), ecc., (c=costante). Infine si utilizzano alcune di esse per determinare delle formule integrali degli L _n ^(α) (x), H_n(x).     

Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m _n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m _n (nα;β,c) as . One holds uniformly for , and the other holds uniformly for , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases. April 16, 1996. Date revised: October 30, 1996.     

On Lp-convergence of Grunwald interpolation

In this paper, the L_p-convergence of Grünwald interpolation G_n(f,x) based on the zeros of Jacobi polynomials J _n ^(α,β) (x)(−1<α,β<1) is considered. L_p-convergence (0<p<2) of Grünwald interpolation G_n(f,x) is proved for p·Max(α,β)<1. Moreover, L_p-convergence (p>0) of G_n(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

Weighted composition operators on growth spaces

Denote by Hol(B _n ) the space of all holomorphic functions in the unit ball B _n of ℂ ⁿ , n ≥ 1. Given g ∈ Hol(B _m ) and a holomorphic mapping φ: B _m → B _n , put C _φ ^g f = g · (f ∘ φ) for f ∈ Hol(B _n ). We characterize those g and φ for which C _φ ^g is a bounded (or compact) operator from the growth space A ^−log(B _n ) or A ^−β (B _n ), β > 0, to the weighted Bergman space A _α ^p (B _m ), 0 < p < ∞, α > −1. We obtain some generalizations of these results and study related integral operators.     

Computations of coefficients in the polynomials of Pade´approximations by solving systems of linear equations

We explore reliability, stability and accuracy of determining the polynomials which define the Pade´approximation to a given function h(x) by solving a system of linear equations to get the coefficients in the denominator polynomial B_n(x). The coefficients in the numerator polynomial A_m(x) follow directly from those for B_n(x). Our approach is in the main heuristic. For the numerics we use the models e^?x1n(1 +x), (1 +x)^{± 1/2} and the exponential integral, each with m=n. The system of equations, with matrix of Toeplitz type, was solved by Gaussian elimination (Crout algorithm) with equilibration and partial pivoting. For each model, the maximum number of incorrect figures in the coefficients is of the order n at least, thus indicating that the matrix becomes ill conditioned as n increases. Let δ_n(x)andω_n(x) be the errors in A_n(x) and B_n(x) respectively, due to errors in the coefficients of B_n(x). For x fixed, δ_n(x) and ω_n(x) and the corresponding relative errors increase as n increases. However, for a considerable range on n, the relative errors in A_n(x)B_n(x) are virtually nil. This has the following theoretical explanation. Now B_n(x)h(x) ?A_m(m) = 0 (x^{m+n+ 1}). It can be shown that ω_n(x)h(x) ? δ_m(x) = 0(x^{m+ 1}). In this sense both A_m(x)B_n(x)andδ_m(x)ω_n(x) are approximations to h(x). Thus if the difference of these two approximations and ω_n(x)B_n(x), the relative error in B_n(x), are sufficiently small, then the relative error in A_m(x)/B_n(x) is of no consequence.     

A note on convergent polynomials of interpolation

In this note we define a sequence {L_n(f;x)} of interpolatory polynomials based on a system x_n={x_kn, k=1,2,…n} of nodes to be a sequence of QLIP if for every f(x)∈C[−1,1], L_n(f; x) tends uniformly to f(x) and ρ_n=1+o(1) as n→∞, where ρ_n is the ratio of the number of points in x_n, at which L_n(f;x) coincides with f(x), and the degree of L_n(f;x). Two sequences of QLIP are constructed, one of which is based on a Bernstein process and the other the Freud-Sharma's construction.     

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ _n f(x) be the sum Σ _{d(x, y)=n} f(y) and letB _n f(x)=Σ _{d(x, y)≦n} f(y). The purpose is to study to what extent Σ _n f andB _n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ _n f orB _n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ _n andB _n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.     

On a problem of (0,2)-interpolation

The object in this paper is to consider the problem of existence, uniqueness, explicit representation of (0,2)-interpolation on the zeros of (1−x²)P_n−1(x)/x when n is odd, where P_n−1 denotes Legendre polynomial of degreen−1, and the problem of convergence of interpolatory polynomials.     

Elliptic equation with singular potential

We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R ⁿ , n ≥ 3:

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

On simultaneous approximation to a differentiable function and its derivative by inverse Pal-Type interpolation polynomials

Let ξ_{n −1} < ξ_{n −2} < ξ_{n − 2} < ... < ξ₁ be the zeros of the the (n−1)-th Legendre polynomial P_n−1(x) and −1=x_n<x_n−1<...<x₁=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C _[−1,1] ¹ , there exists a unique polynomial R_n(x) of degree 2n−2 (if n is even) satisfying conditions R_n(f, ξ_k) = f (ε_k) (1 ⩽ k ⩽ n −1); R¹ _n(f,x_k)=f¹(x_k)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {R_n(f, x)} (n is even) and the main result of this paper is that if f∈C _[1,1] ^r , r≥2, n≥r+2, and n is even then |R¹ _n(f,x)−f¹(x)|=0(1)|W_n(x)|h(x)·n^3−r·E_2n−r−3(f^(r)) holds uniformly for all x∈[−1,1], where .     

LIMIT OF ITERATES FOR BERNSTEIN POLYNOMIALS DEFINED ON A TRIANGLE

Let f∈C[0,1],and Bn(f,x) be the a-th Bernstein polynomial associated with function f.ln 1967,the limit of iterates for B.(f,x) was given by Kelisky and Rivlin.After this,Many mathematicians studied and generalized this result.But anyway,all these discussions are only for univariate case ,In this paper,the main contrlbution is that the limit of lterates for Bernstein polynomial defined on a triangle is given completely.     

Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant α(Q ² /Λ ² ) = β ₁ α_s(Q ² )/(4π) becomes a Q ² -analytic invariant function α _an (Q²/Λ ² ) ≡A(x), which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable F, instead of powers of the analytic invariant charge A(x), may contain specific functions A_n(x)=[aⁿ(x)] _an , the “nth power of a(x) analyticized as a whole.” Functions A _n>2(x) for small Q² ≤Λ ² oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for F(x) becomes an asymptotic expansion à la Erdélyi using a nonpower set {A _n (x)}. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 55–66, April, 1999.     

Growth of the coefficient of quasiconformality and the boundary correspondence of automorphisms of a ball

A homeomorphismf:B ⁿ→B ⁿ of the unit ball inR ⁿ(n≥2) whose coefficient of quasiconformality in the ball of radiusr<1 has asymptotic rate of growthK(r)=sup _|x|≤r k(x, f)=O(log (1/1−r)) can be continued to a homeomorphism of the closed ball . Forn=2 this implies that the Caratheodory theory of prime ends for conformal mappings also holds for the class of homeomorphismsf:B ²→D withK(r)=O(log (1/1−r)). This work was partially supported by SIZ za nauku SRCG, Titograd.     

A family of Bernstein quasi-interpolants on [0,1]

Suppose that we want to approximate f∈C[0,1] by polynomials inP, using only its values on X_n={i/n, 0≤i≤n}. This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f. But, when n tends to infinity, L_n f does not converge to f in general and the convergence of B_n f to f is very slow. We define a family of operators B _n ^(k) , n≥k, which are intermediate ones between B _n ⁽⁰⁾ =B _n ⁽¹⁾ =B_n and B _n ⁽ⁿ⁾ =L_n, and we study some of their properties. In particular, we prove a Voronovskaja-type theorem which asserts that B _n ^(k) f−f=O(n^−[(k+2)/2]) for f sufficiently regular. Moreover, B _n ^(k) f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.     

Some questions in the theory of inverse problems for differential operators

For the two operatorsLy=y ⁿ +σ _k=0 ⁿ⁻² p _k (x)y( ^k ) and Ry=yⁿ+σ _k=0 ⁿ⁻² p_k(x)y(^k) with a common set of boundary conditions we establish a connection between p_k(x) and P_k(x) in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This enables us to recover the operator L from the operator R by solving a system of nonlinear ordinary differential equations. For Sturm-Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 151–160, February, 1977. I wish to express my thanks to my scientific adviser V. A. Sadovnich.