Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.     

On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL ₁=L₁(0,2π) 2π-перио дических функций кла сса W^rH^ω = {f:f∈C^r,ω(f ^(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W ₁ ^r+v N = {?: ?^r+v?1)(t) —локально абсолютн о непрерывна, ∥?(^r+v∥L₁≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W ₁ ^r+v в L₁, обладает аналог ичным свойством относител ьно класса W^rH^ω при любом выпуклом вверх ω(δ).     

On approximation of convex functions by rational ones in integral metrics

Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, R_n(f,p Δ) = inf∥f?r∥_Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {R_n(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|^1/pn^?2,где С(р) - величина, зави сящая лишь от р.     

Algebraic independence of the values of certain series by Mahler's method

Suppose thatf ₁(z), ...f _m(z) are algebraically independent functions of a complex variable satisfying $$f_i (z) = a_i (z)f_i (Tz) + b_i (z),$$ wherea _i (z),b _i (z) are rational functions andTz=p(z ^?1)^?1 for a polynomialp(z) of degree larger than 1. We show thatf ₁(a), ...,f _m (a) are algebraically independent under suitable conditions onf anda. As an application of our main result, we deduce three corollaries, which generalize earlier work by Davison and Shallit and by Tamura.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

A boundedness theorem for a certain nth order differential equation

Summary Consider the equation(1.2) in which a₁, a₂, ..., a_n−1 are constants and the functions f_n(x) and p(t) (both continuous) together with are all bodnded. A recent investigation by Reissig[1] shows that if f_n(x)sgn x>0 (|x|≥h>0) then subject to certain conditions, which are stated explicitly in[1], the solutions of such an equation(1.2) are all ultimately bounded The object of the psesent paper is to generalize that result to the equation(1.1) in which φ_n is a bounded function depending on all the variables shown, and each coefficient ϕ_i (i=2,3,..., n−1) satisfies as |ζ|→∞ for some constant a_i. Entrata in Redazione il 20 agosto 1970.     

The average quality of greedy-algorithms for the Subset-Sum-Maximization Problem

This paper deals with the quality of approximative solutions for the Subset-Sum-Maximization-Problem maximize $$\sum\limits_{i = l}^n {a_i x_i } $$ subject to $$\sum\limits_{i = l}^n {a_i x_i } \leqslant b$$ wherea _l,...,a_n,bεR⁺ andx _l,...x_nε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x ₁, ..., x_n) whose objective value is less or equal the optimal value, which is of course not greater thanb. We use the gap between capacityb and realized value as an upper bound for the error made by the heuristic and as a criterion for quality. Under the stochastic model:a ₁, ..., a_n, b independent,a ₁...,a_n uniformly distributed on [0, 1], b uniformly distributed on [0,n] we derive the gap-distributions and the expected size of the gaps. The analyzed algorithms include four algorithms which can be done in linear time and four heuristics which require sorting, which means that they are done inO(nlnn) time.     

Stability of solutions of functional equations connected with problems of characterizing probability distributions

The work contains some results pertaining to the solution ψ_j(x) of the functional equation $$\left| {\Sigma \Psi _j \left( {a_j^T t} \right)} \right| \leqslant \varepsilon ,$$ where a _j ^T =(a_1j, a_2j, ..., a_pj)∈ ?^p, all the coefficients a_ij are constant, t=(t₁, t₂, ..., t_p) ∈ ?^p, \(a_j^T t = \sum\limits_{i - 1}^p {a_{ij} t_i } ,p \geqslant 2\) and the relation (*) is satisfied for all Inequality (*) is connected with certain characterization theorems of probability theory and statistics. For simplicity, it is assumed that the ψ_j(x) are continuous functions, x∈?¹. The following basic ressult is obtained.     

Solvability of vector integro-differential equations of convolution type on the semiaxis

The paper deals with vector integro-differential equation of convolution type that have the form

$ - \frac{{d^2 f_i }}{{dx^2 }} + a_i f_i (x) = g_i (x) + \sum\limits_{j = 1}^N {\int\limits_0^\infty {K_{ij} (x - t)f_j (t)dt, } i = 1,2, \ldots ,N,} $

where \(\vec f\) = (f ₁, f ₂, ..., f _N ) ^T is the unknown vector-function, a _i are nonnegative numbers, \(\vec g\) = (g ₁, g ₂, ..., g _N ) ^T ? L ₁ ^×N (0,+∞) ≡ L ₁ (0,+∞) × ... × L (0,+∞) is the independent term of the equation with nonnegative components and 0 ≤ K _ij ? L ₁ (?∞,+∞), i, j = 1, 2, …,N are the kernel-functions. These equations have significant applications in the wave non-local interaction theory. Using some special factorization methods, solvability of the system is proved in different functional spaces.

     

Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa ₁,a ₂, ...,a _r and denominator parametersb ₂, ...,b _r, we say it isalmost poised ifb _i, =a ₁ q ^δ,i a _i,δ_i = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a₁, =q ⁻²ⁿ. Partially supported by N.S.F. Grant No. DMS-8521580.     

The uniqueness of the element of best mean approximation to a continuous function using splines with fixed nodes

Suppose that on the Interval [a, b] the nodes $$a = x_0< x_1< \ldots< x_m< x_{m + 1} = b$$ are given and the functions u₀(t)=ω₀(t) $$u_i (t) = \omega _0 (t)\smallint _0^t \omega _1 (\varepsilon _1 )d\varepsilon _1 \ldots \smallint _a^{\varepsilon _{\iota - 1} } \omega _1 (\varepsilon _1 )d\varepsilon _\iota ,\varepsilon _0 = t(i = 1,2, \ldots ,n)$$ where the functions ω_i(t)> 0 have continuous (n?i)-th derivatives (i=0, 1, ..., n). S_n,m will designate the subspace of functions that have continuous (n?1)-st derivatives on [a, b] and coincide on each of the intervals [x_j, x_j+1] (j=0, 1, ..., m) with some polynomial from the system {u_i(t)} _i=0 ⁿ .THEOREM. For every continuous function on [a, b] there exists in S_n,m a unique element of best mean approximation.     

Estimate of a complete rational trigonometric sum

Supposef is a polynomial of degree n≥3 with integral coefficientsa ₀,a ₁,...,a _n; q is a natural number; (a ₁,...,a _n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ .     

Approximation of functions of finite variation by superpositions of a Sigmoidal function

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ^d ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝ^d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L₂-norm by elements of the set G_n = {Σ_i=0^staggeredn c_ig(〈a_i, x〉 + b_i) : a_iℝ^d, b_i, c_iℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n^1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ^d6t6e^d|u(t)| > ∞     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Estimate of a sum of Legendre symbols of polynomials of even degree

Let n≥4 be even, p > (n²?2n)/2 be simple odd, andf(x)=a ₀+a ₁+...+a _nxⁿ be a polynomial with integral coefficients that are not quadratic over the residue field modulo p, (a _n, p)=1. The following inequality is proved: $$\left| {\sum\nolimits_{x = 1}^p {\left( {\frac{{f(x)}}{p}} \right)} } \right| \leqslant (n - 2)\sqrt {p + 1 - \frac{{n(n - 4)}}{4}} + 1.$$      

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Cantor sets arising from continued radicals

If a ₁,a ₂,a ₃,… are nonnegative real numbers and $f_{j}(x) = \sqrt{a_{j}+x}$ , then lim _n→∞ f ₁°f ₂°?°f _n (0) is a continued radical with terms a ₁,a ₂,a ₃,…. The set of real numbers representable as a continued radical whose terms a _i are all from a set S={a,b} of two natural numbers is a Cantor set. We investigate the thickness, measure, and sums of such Cantor sets.     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Singular series of genus 2 for sums of squares

Let f₁, ..., f_h be representatives of classes in the genus of a positive quadratic form f, r_i(k) the number of representations of k by the form f_i, (k) the value of the singular Hardy-Littlewood series, E_i the number of integer automorphisms of f_i. We derive an expression for in terms of the values of singular series of genus 1 and 2. This expression is evaluated for the simplest model f(x)=x ₁ ² +...+x _m ² , m1 (mod 8), k is a prime in-integer, k=1(mod 8).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 26–39, 1986.