Approximating periodic functions in the uniform metric by Jackson type polynomials

Let C be the space of continuous 2π-periodic functions f with the norm . Let , where , be the Jackson polynomials of a function f, E _n (f) be the best approximation of f in the space C by trigonometric polynomials of order n, and let , be the function trigonometrically conjugate to the primitive of f. The paper establishes results of the following types:

Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let

Approximating periodic functions by interpolation sums of Jackson type

Let

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

An error bound of the Ritz method for a singular second-order differential equation

The paper presents an error bound of the Ritz method for the problem of minimizing the functional

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

The area of an exponential random walk and partial sums of uniform order statistics

Let S_i be a random walk with standard exponential increments. The sum ∑ _i=1 ^k S_i is called the k-step area of the walk. The random variable ∑ _i=1 ^k S_i plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that

Caratterizzazione delle tracce per spazi con peso

Sunto Si prova un teorema di tracce per spazi di funzioni, definite su R ₊ ⁿ , con norme del tipo . Entrata in Redazione il 23 luglio 1969. Lavoro eseguito nell'ambito dei gruppi di ricerca del Comitato Nazionale per la Matematica del C.N.R.     

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

Littlewood-Paley theorem for arbitrary intervals: Weighted estimates

Let 1 < r < 2 and let b is a weight on ℝ such that satisfies the Muckenhoupt condition A_r′/2 (r′ is the exponent conjugate to r). If f_j are functions whose Fourier transforms are supported on mutually disjoint intervals, then

Equivalent definition of some weighted Hardy spaces

We present an equivalent definition of functions analytic in the half-plane ℂ₊ = {z: Re z > 0} for which

On extremal properties for the polar derivative of polynomials

If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.