Best quadrature formulas for improper integrals on certain classes of differentiable functions

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 132–134, June, 1991.     

Best quadrature formulas on classes of differentiable periodic functions

A solution is given to the problem of finding the best quadrature formula among formulas of the form $$\int_0^{2\pi } {f(x)dx \approx \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {pk,l} f^{(l)} (x_k ),} } $$ which are exact in the case of a constant, for p = r ? 1, r = 1, 2, 3... and p = r ? 2, r even, for the classes W(r) LqM of 2π-periodic functions.     

On inequalities for seminorms of certain classes of differentiable periodic functions

We obtained the inequalities for upper bounds of seminorms of classes of 2π-periodic functions, which are determined by a linear differential operator and by the majorant of the modulus of continuity.     

Approximation of certain classes of differentiable functions by generalized splines

We find the exnet value of the best (α, β)-approximation by generalized Chebyshev splines for a class of functions differentiable with weight on [?1, 1].     

Best one-sided approximation of certain classes of functions

This considers the question of the best one-sided approximation of certain classes of continuous periodic functions by means of trigonometric polynomials of order n-1 in the metric L ₂ ^p (1p<). Precise upper bounds are obtained for the best one-sided approximation of classes of 2/n-periodic functions H_,n [having arbitrary prescribed modulus of continuity (t)] in the metric L ₂ ^p , as well as of classes of 2-periodic functions H [having prescribed modulus of continuity (t) with definite limits] in the metric L ₂ ¹ .Translated from Matematicheskie Zametki, Vol. 10, No. 6, pp. 615–626, December, 1971.The author warmly thanks N. P. Korneichuk for his constant attention to this work.     

Optimal recovery methods for the integral on classes of differentiable functions

Найдены методы восст ановления интеграла по информации $$I\left( f \right) = \left\{ {f^{(j)} \left( {x_i } \right)\left( {j = 0, ..., \gamma _i - 1; i = 1, ..., n; 1 \leqq \gamma _i \leqq r; \gamma _i + ... + \gamma _n \leqq N} \right.} \right\},$$ оптимальные на класс ахW _p ^r ,r=1,2,...; 1≦p≦∞. Это позволило, в частност и, получить наилучшие для классаW _p ^r квадратурные форму лы вида $$\mathop \smallint \limits_0^1 f\left( x \right)dx = \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 1}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right)$$ И $$\mathop \smallint \limits_0^1 f\left( x \right)dx = af\left( 0 \right) + \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 0}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + bf\left( 1 \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right).$$      

Approximation of classes of differentiable functions

Conditions are derived under which is finite or infinite. The value of F is calculated for certain special cases.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 105–112, February, 1971.     

The exact error of trigonometric interpolation for differentiable functions

In [1], G. Halász gives some properties of the order of trigonometric approximation as a function of the Lipschitz parameter. Here we show that these properties completely characterize this function.     

On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

Approximation of certain classes of differentiable periodic functions by interpolational splines in a uniform decomposition

In this paper we obtain upper and lower bounds of a uniform approximation by interpolational splines of order r in a uniform decomposition on the classes of functions W^rH_ω and W _L ^r+2 and on the whole space C^r.