A note on the non-commutative neutrix product of distributions

The distributionF(x ₊, −r) Inx₊ andF(x ₋, −s) corresponding to the functionsx ₊ ^−r lnx₊ andx ₋ ^−s respectively are defined by the equations

On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

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.     

The rate of convergence for subexponential distributions

A distribution functionF on the nonnegative real line is called subexponential if

On the convergence of optimum stratifications for empiric distribution function in univariate case

Summary Let {X _n },n=1,2,..., be a sequence of independent random variables distributed according to a distribution functionF(x) with finite variance,F _n (x) be the empiric distribution function ofX ₁,...,X _n for eachn, andφ _(n) ^* andφ ^* be optimum stratifications corresponding toF _n (x) andF(x) respectively. It is shown in this paper thatφ _(a) ^* tends almost surely toφ ^* under a suitable criterion. Institute of Statistical Mathematics     

Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S ⁿ :x. y ≥ 0},x ∃S ⁿ, whereS ⁿ is thendimensional unit sphere in ℝⁿ⁺¹,n ≥ 2, and Μ is a finite Borel measure onS ⁿ. If Μ is absolutely continuous with respect to Lebesgue measuredy onS ⁿ, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫ _{x.y> 0} f(y)dy and consider the following cases: (a)f ∃C ^∞(Sⁿ); (b)f ∃ L^p(S ⁿ), 1 ≤ p < ∞; and (c)f ∃C(Sⁿ). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL _p ^γ (Sⁿ) is studied. For zonalf ∃ L¹(S ²), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation

We consider the computation of the Cauchy principal value integral by quadrature formulae Q _n ^F [f] of compound type, which are obtained by replacing f by a piecewise defined function F_n[f]. The behaviour of the constants k_{i, n} in the estimates [R _n ^F [f]] |⩽K _i,n ‖f ⁽ⁱ⁾‖_∞ (where R _n ^F [f] is the quadrature error) is determined for fixed i and n→∞, which means that not only the order, but also the coefficient of the main term of k_{i, n} is determined. The behaviour of these error constants k_{i, n} is compared with the corresponding ones obtained for the method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.     

Characterization of certain classes of orthogonal polynomials related to elliptic functions

Summary In this paper we determine all orthogonal polynomials U_n(x) such that U_n(x)=x^−1/2 F _2n+1 (x ^1/2 ) and where f(t), u(t) have Taylor series expansions. Supported in part by N. S. F. grant GP-1593.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

Generalizations of Non-Commutative Neutrix Convolution Products of Functions

The non-commutative neutrix convolution product of the functions x ^r cos_–(x) and x ^s cos₊(x) is evaluated. Further similar non-commutative neutrix convolution products are evaluated and deduced.     

On the regular growth of Dirichlet series absolutely convergent in a half-plane

For the Dirichlet series F(s) = ?_{n = 1}^￥ a_nexp{ sl_n } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ _a =0, we establish conditions for (λ _n ) and (a _n ) under which lnM( s, F ) = T_R( 1 + o(1) )exp{ r_R

Piecewise Monotone Pointwise Approximation

Let r, k, s be three integers such that , or We prove the following: Proposition. Let Y:={y _i } _i=1 ^s be a fixed collection of distinct points y _i ∈ (-1,1) and Π (x):= (x-y ₁ ). ... .(x-y _s ). Let I:=[-1,1]. If f ∈ C ^(r) (I) and f'(x)Π(x) ≥ 0, x ∈ I, then for each integer n ≥ k+r-1 there is an algebraic polynomial P _n =P _n (x) of degree ≤ n such that P _n '(x) Π (x) ≥ 0 and $$ \vert f(x)-P_n(x) \vert \le B\left(\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)^r \omega_k \left(f^{(r)};\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right) \legno{(1)}$$ for all x∈ I, where ω _k (f ^(r) ;t) is the modulus of smoothness of the k -th order of the function f ^(r) and B is a constant depending only on r , k , and Y. If s=1, the constant B does not depend on Y except in the case (r=1, k=3). In addition it is shown that (1) does not hold for r=1, k>3. March 20, 1995. Dates revised: March 11, 1996; December 20, 1996; and August 7, 1997.     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

Spectral invariance for algebras of pseudodifferential operators on besov-triebel-lizorkin spaces

The algebra of pseudodifferential operators with symbols inS _1,δ ⁰ , δ<1, is shown to be a spectrally invariant subalgebra of ℒ(b _p,q ^s ) and ℒ(F _p,q ^s ). The spectrum of each of these pseudodifferential operators acting onB _p,q ^s orF _p,q ^s is independent of the choice ofs, p, andq.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.     

Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that ?_{n ￡ x} f(n)=o(x)\sum_{n\le x} f(n)=o(x) . We propose sufficient conditions on the function f to deduce the equality ?_{n ￡ x} f(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x) . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

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.