Приближение непреры вных функций средним и Эйлера

Let _n(1,f,x)=1/2 ⁿ _k=1 ⁿ C _k ⁿ S_k(x,f) denote the Euler means of the Fourier series of the 2-periodic functionf(x). For a function the main term of deviationf(x)– _n (1,f, x) is calculated. Asymptotically exact order of decrease of the upper bound of such deviations over the classH ⁽⁾ is also obtained.     

Pointwise estimation of comonotone approximation

We prove that, for a continuous functionf(x) defined on the interval [–1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP _n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)–P _n (x) ¦C₂(f;n^–2+n ^–11–x ²), whereC is a constant that depends on the length of the smallest interval.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.     

Piecewise polynomial approximation

For best piecewise polynomial approximation _n=_n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation _n=o[ _f (e ^–n )] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972.     

On the convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf ¹ _–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp _n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].     

On the extreme spectral properties of Toeplitz matrices generated byL 1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue ₁ ⁽ⁿ⁾ of symmetric (Hermitian)n ×n Toeplitz matricesT _n (f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that ₁ ⁽ⁿ⁾ tends to essinff =m _f in the following way: ₁ ⁽ⁿ⁾ –m _f 1/n ^2k . These authors use three assumptions:A1)f –m _f has a zero inx =x ₀ of order 2k.A2)f is continuous and at leastC ^2k in a neighborhood ofx ₀.A3)x =x ₀ is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L ¹[–, ]. In this paper we further extend this theory to the case of a functionf L ¹[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m _f is the only parameter which characterizes the rate of convergence of ₁ ⁽ⁿ⁾ tom _f .     

Approximation by linear methods of summation of orthogonal series

We give an estimate of the rapidity of convergence of certain linear means of the orthogonal series _k =0 c_{k k} (x) to the functionf (x) L₂ (a, b) defined by this series according to the Riesz-Fisher theorem for almost all x (a, b). The results obtained are, in a certain sense, final.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 345–356, March, 1968.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

On the order of approximation of a continuous 2π-periodic function by Fejer and Poisson means of its Fourier series

Let _n (f) and P_r (f) be, respectively, the Fejer and Poisson means of the Fourier series of the functionf. The present work considers problems associated with the rapidity of approximation of a continuous 2-periodic function by means of Fejer and Poisson processes, and gives, in particular, an upper bound to the deviation of the Fejer and Poisson processes from the function in terms of moduli of continuity, and a lower bound to _n (f)–f in terms of functionals composed of best approximations to the functionf; in addition, some relationships among the quantities P_r (f)–f and _n (f)–f are established.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 21–32, July, 1968.     

The approximation of functions of many variables by their Féjér sums

We construct elliptic Féjér polynomials K_n(x) of m variables. We prove some of their properties: a) the Féjér polynomials are positive on the m-dimensional torus T^m, K_n(x)0, b) (x)=o(n^–1), as n, c) we calculate their norms in the spaces L[T^m] and C[T^m]. We estimate the deviation of the Féjér sum _n(x,f) from the functionf(x). For the space C[T^m]: where c_,m c_1,m are constants.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 817–828, June, 1973.In conclusion, the author wishes to express his gratitude to S. B. Stechkin for help with the paper.     

The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

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.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

On the greatest length of a dead-end disjunctive normal form for almost all Boolean functions

The maximal numberl(f) of conjunctions in a dead-end disjunctive normal form (d.n.f.) of a Boolean functionf and the number (f) of dead-end d.n.f. are important parameters characterizing the complexity of algorithms for finding minimal d.n.f. It is shown that for almost all Boolean functionsl(f)2^n–1, log₂ (f)2^n–1log₂nlog₂log₂n (n).Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 649–658, December, 1968.     

Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Eigenfunctions and associated functions of an n-th-order linear differential operator

For n2 we consider a differential operatorL [y] z ⁿ y ⁽ⁿ⁾ +P ₁(z)z ^n–1 y ^(n–1) +P ₂ (z)z ^n–2 y ^n–2 + ...+P _n (z)y = y, p ₁ (z), ..., P _n (z) A _R : here a_r is the space of functions which are analytic in the disk ¦z¦ < R, equipped with the topology of compact convergence. We prove the existence of sequences {f_k(z)} _k =o, consisting of a finite number of associated functions of the operator L and an infinite number of its eigenfunctions; we show that the sequence forms a basis in A_r for an arbitrary r, 0 < r <- R; and we establish some additional properties of the sequence ₀ (z), ₁ (z),..., _d–1 (z), f _d (z), f _d+1 (z),... Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 869–878, December, 1976.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Imbedding in the class eL

Conditions which must be satisfied by the modulus of continuity and smoothness of a functionf(x) L_p(0, 2) in order thatf(x) or (x) belong to the class e^L are obtained.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 17–24, July, 1971.     

An extreme value theory for long head runs

Summary For an infinite sequence of independent coin tosses withP(Heads)=p(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1–p)–log(k!)+k+/–1/2+ r₁(n)+ o(1) where log=log_1/p , =0.577 ... is the Euler-Mascheroni constant, =ln(1/p), andr ₁(n) is small. The variance is ₂/6²+1/12 +r ₂(n)+ o(1), wherer ₂(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.This work was supported by a grant from the System Development Foundation     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part II: Clenshaw–Lord Type Approximants

Laurent–Padé (Chebyshev) rational approximants P _m (w,w ^–1)/Q _n (w,w ^–1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P _m /Q _n matches that of a given function f(w,w ^–1) up to terms of order w ^±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ^±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P _m matches that of Q _n f up to terms of order w ^±(m+n), but based on knowledge of the series coefficients of f up to terms in w ^±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).