Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [? π, π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y:= {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that, on [y _i , y _i?1], f is nondecreasing if i is odd and nonincreasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₂(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator B_r,r(f) which maps the set of 2-periodic functionsf(t) X^(r) (X^(r)=C^(r) or L₁ ^(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L₁ metric, E_n(f)_X and (f, )_X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; K_r are the known Favard constants.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 21–30, July, 1973.In conclusion, the author wishes to express his deep gratitude to N. P. Korneichuk under whose guidance this paper was written.     

On series arising from the approximation of periodic differentiable functions by Poisson integrals

Mathematical Notes - For series of special form, we obtain an expansion in powers of the parameter. The coefficients of the expansion can be written out explicitly in terms of Bernoulli...     

Classification of infinitely differentiable periodic functions

The set of infinitely differentiable periodic functions is studied in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. In particular, we establish that every function f from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function. At the same time, for an arbitrary function f ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. We also obtain new criteria for 2π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions. Deceased. (A. I. Stepanets) Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1686–1708, December, 2008.     

Approximation of weakly differentiable periodic functions

An asymptotic equality is found for the lower bounds of the best approximations of the classes C , under a condition of slow growth of (·).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 406–412, March, 1990.     

On the order of relative approximation of classes of differentiable periodic functions by splines

In the case where n → ∞, we obtain order equalities for the best L _q -approximations of the classes W _p ^r , 1 ≤ q ≤ p ≤ 2, of differentiable periodical functions by splines from these classes.     

The approximation of periodic differentiable functions by splines with respect to a uniform partition

We solve the problem of determining exact estimates for the approximation by r-th order splines of the class W^r+1 in the metrics C and L_p (1p<).Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 807–816, June, 1973.     

On best approximation of infinitely differentiable functions

Dnepropetrovsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 5, pp. 12–28, September–October, 1991.     

Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions

The author examines matrices and bordered matrices having exactly one or at most one negative eigenvalue. These results are then used to state matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. For quadratic functions, a new criterion for quasiconvexity is established, and its equivalence to the already known criterion [11,23] is also shown.     

Linear method for the approximation of differentiable functions

This article is devoted to the problem of the approximation of functions in the metric of space L_p(0, 1), the s-th derivative of the functions being continuous and the (s + l)-th derivative belonging to the space L_q(0, 1), with p, q 1, by spline functions of order s with fixed almost uniform nodes.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 423–430, April, 1970.     

Coconvex approximation of periodic functions

The Jackson inequality relates the value of the best uniform approximation E _n (f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω ₃(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials coconvex to them. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007.     

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and