Bounds from below on diameters of classes of periodic functions with a bounded fractional derivative

Shevaldin, V. T.

doi:10.1007/bf01208331

Cited by 2 publications

(6 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The spline SΨ β,1 (y, ·) generates a system of fundamental splines of the form SΨ β,1 (y, · − x k ), k = 0, 2n − 1, and this system represents a basis in the space SΨ β,1 (∆ 2n ). Necessary and sufficient conditions of existence and uniqueness for the fundamental spline SΨ β,1 (y, ·), which depends on the relation between y (it is a displacement of interpolation nodes) and parameters ψ and β of the generator kernel Ψ β,1 , were studied in [6,10,11,14,15,20].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Equalities in (13) and (14) are understood as the equality of two functions in L (i.e., almost everywhere). Due to Lemma 2.3.4 from [3, p. 76], the function in the right-hand side of equality (14) is constant on each interval (x k , x k+1 ). Therefore, among (ψ, β)-derivatives of any spline of the form (12), and hence, for the fundamental spline SΨ β,1 (·) as well, one can find a function which is constant on each interval (x k , x k+1 ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exact values of Kolmogorov widths of classes of Poisson integrals

Serdyuk

Bodenchuk

2013

Journal of Approximation Theory

View full text Add to dashboard Cite

We prove that the Poisson kernel $P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos(kt-\dfrac{\beta\pi}{2})$, ${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with a number $n_q$ where $n_q$ is the smallest number $n\geq9$, for which the following inequality is satisfied: $$ \dfrac{43}{10(1-q)}q^{\sqrt{n}}+\dfrac{160}{57(n-\sqrt{n})}\; \dfrac{q}{(1-q)^2}\leq (\dfrac{1}{2}+\dfrac{2q}{(1+q^2)(1-q)})(\dfrac{1-q}{1+q})^{\frac {4}{1-q^2}}. $$ As a consequence, for all $n\geq n_q$ we obtain lower bounds for Kolmogorov widths in the space $C$ of classes $C_{\beta,\infty}^q$ of Poisson integrals of functions that belong to the unit ball in the space $L_\infty$. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes $C_{\beta,\infty}^q$ and show that subspaces of trigonometric polynomials of order $n-1$ are optimal for widths of dimension $2n$

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact values of Kolmogorov widths of classes of Poisson integrals

Serdyuk

Bodenchuk

2013

Journal of Approximation Theory

View full text Add to dashboard Cite

show abstract

“…In particular, it was proved in [35,36] that, for β ∈ Z, the condition C y n ,2 , n ∈ N , is satisfied by functions Ψ β ( ) t of the form (55) with coefficients ψ ( k ) = ϕ ρ ( ) k k , 1 < ρ ≤ 1 / 7 , where ϕ ( k ) are arbitrary positive nonincreasing functions of natural argument. Later, Shevaldin proved in [37,38] that the condition C y n It was proved in [20] that the conditions C y n ,2 , n ∈ N , are also satisfied by kernels Ψ β ( ) t of the form (5) whose coefficients ψ ( k ) satisfy the inequalities…”

Section: Estimates For Kolmogorov Widths Of Classesmentioning

confidence: 97%

“…where W t n ( ) = q n q jt R t n n j j n n j n sin cos In this case, with regard for estimates (38), (40), and (50), we get…”

Section: Propositionmentioning

confidence: 98%

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Serdyuk

2005

Ukr Math J

View full text Add to dashboard Cite

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ β whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L.Let L = L 1 be the space of 2π-periodic functions ϕ summable on [ 0, 2 π ] with the normthe space of measurable, essentially bounded, 2π-periodic functions ϕ with the norm ϕ ∞ = ess sup t t ϕ ( ) , and let C be the space of continuous 2π-periodic functions ϕ with the norm ϕ C = max t t ϕ ( ) .In the present paper, we use classes of functions introduced by Stepanets [1], namely, the classes L β ψ ᑨ,

show abstract

Bounds from below on diameters of classes of periodic functions with a bounded fractional derivative

Cited by 2 publications

References 4 publications

Exact values of Kolmogorov widths of classes of Poisson integrals

Exact values of Kolmogorov widths of classes of Poisson integrals

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

Contact Info

Product

Resources

About