On the approximation of periodic functions by de la Vallée Poussin sums

Пустьf — непрерывная периодическая функц ия,s _n (f) — сумма Фурье порядкаn функцииf,E _n (f) — наилучшее прибли жениеf тригонометри ческими полиномами порядкаn в чебьппев-ской метрике и $$\sigma _{n, m} (f) = \frac{1}{{m + 1}}\mathop \sum \limits_{v = n - m}^n s_v (f) (0 \leqq m \leqq n; n = 0, 1, \ldots )$$ — суммы Bалле Пуссена ф ункцииf Для любой последовательностиε={ε_v} (v=0, l,...),ε _v ↓0(v→∞) обозначим чер езC(ε) класс непрерывн ых функцийf, для которыхE _v (f)≦ε _v (v=0,1,...). В работе устанавли вается, что существую т абсолютные положите льные кон-стантыa ₁ иa ₂ такие, что $$A_1 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}} \leqq \mathop {\sup }\limits_{f \in C(\varepsilon )} \parallel f - \sigma _{n, m} (f)\parallel \leqq A_2 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}}$$ для всех 0≦m≦n; n=0, l, ... В частн ых случаяхт=п иm=0 этот результат равноси-ле н теоремам, установлен ным ранее автором и К. И. Осколковым.     

Asymptotics of the approximation of continuous and differentiable functions by the singular integrals of de la Vallée Poussin

The complete asymptotic developments in powers of 1/n are derived for quantities characterizing approximation by singular integrals of de la Vallée Poussin $$V_n (f:x) = \frac{1}{{\Delta _n }}\int_{ - \pi }^\pi {f(x + t)} \cos ^{2n} \frac{t}{2}dt;\Delta _n = \int_{ - \pi }^\pi {\cos ^{2n} \frac{t}{2}dt}$$ of the function classes Lipa, 0w ^(r), r?1 an integer.     

On the degree of approximation by the operators of de la Vallée Poussin

In this paper we study the degree of approximation of functionsf inC ₂ andC ₂ ¹ by the operatorsV _n ofde la Vallée Poussin. The quality of approximation is measured in terms of the modulus of continuity off andf respectively. Forn so-called exact constants of approximation are determined. Furthermore, the asymptotic behaviour of these constants is investigated asn.     

On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

Best approximation and de la Vallée-Poussin sums

For the class C_ε={f∈C_2π: E_n, n≤Z₊} where \(\left\{ {\varepsilon _n } \right\}_{n \in Z_ + } \) is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée-Poussin sums: (1) $$c_1 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \leqslant \mathop {\sup }\limits_{f \in C_\varepsilon } \left\| {f - V_{n, l} \left( f \right)} \right\|_C \leqslant c_2 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \left( {n \in N} \right)$$ , where c₁ and c₂ are constants which do not depend on n orl. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes C_ε of (differentiable) functions. The result (1) substantially refines the estimate (see [1]) (2) $$\left\| {V_{n, l} \left( f \right) - f} \right\|_C = O\left( {\log {n \mathord{\left/ {\vphantom {n {\left( {l + 1} \right) + 1}}} \right. \kern-\nulldelimiterspace} {\left( {l + 1} \right) + 1}}} \right) E_n \left[ f \right] \left( {n \to \infty } \right)$$ and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]).     

Uniform approximation on [?1, 1] via discrete de la Vallée Poussin means

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss?CJacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.     

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

Approximation of classes of continuous functions by generalized de la Vallée-Poussin sums

We introduce generalized de la Vallée-Poussin sums and study their approximation properties for the classes of continuous periodic functionsC _, .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1069–1079, August, 1995.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

.     

Qp空间中的de la Vallée Poussin平均

本文研究了单位球上的Q_p空间中的de la Vallée Poussin平均算子,并通过高阶光滑模来建立Jackson逼近定理.此外,我们还得到了Bernstein不等式,K-泛函和光滑模的等价刻画等结果.     

Über eine Interplationsformel von de la Vallée Poussin

Ohne Zusammenfassung     

Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces L _s , 1 ≤ s ≤ ∞, we establish asymptotic equalities for upper bounds of approximations by de la Vallée-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallée-Poussin sums in the metrics of the spaces L _s , 1 ≤ s ≤ ∞, on the classes of Poisson integrals of functions belonging to the unit ball of the space L ₁.     

Approximations of functions defined on the real axis by means of de la Vallée-Poussin operators

Asymptotic equations for upper bounds of deviations of the de la Vallée-Poussin operators on C classes in the uniform metric are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 682–691, May, 1992.     

Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

One considers linear summation methods for the multiple Fourier series the multidimensional analogues of the de la Vallé-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron . It is shown that if has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)^–1. log^m–1 (n+2)) are equal to where is the Fourier transform of the function . The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) =0. is an m-dimensional simplex.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 154–165, 1983.