Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

Abdullayev, G Fahreddin; Özkartepe, P.; Savchuk, V. V.; Shidlich, Andrii

doi:10.2298/fil1905471a

Cited by 12 publications

(13 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the spaces L 2 of 2π-periodic square-summable functions, for moduli of continuity, this result was obtained by A.G. Babenko [4]. In the spaces S p of functions of one and several variables, this result for classical moduli of smoothness was obtained in [28] and [2], respectively, and for generalized moduli of smoothness, in [1] (for functions of one variable). In the proof of Theorem 1, we mainly use the ideas outlined in [4,14,15,28], taking into account the peculiarities of the spaces BS p .…”

Section: Jackson Type Inequalitiesmentioning

confidence: 86%

See 1 more Smart Citation

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Serdyuk

Shidlich

2021

Carpathian Math. Publ.

View full text Add to dashboard Cite

show abstract

Section: Jackson Type Inequalitiesmentioning

confidence: 86%

“…functions. In the spaces S p of functions of one and several variables, Theorem 1 and Corollaries 1, 3 and 4 were proved in [28] and [2], respectively. In the spaces L 2 , for classical moduli of smoothness inequality (19) was proved by N.I.…”

Section: Jackson Type Inequalitiesmentioning

confidence: 98%

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Serdyuk

Shidlich

2021

Carpathian Math. Publ.

View full text Add to dashboard Cite

show abstract

“…The results obtained in this direction are formulated in the form of so-called direct and inverse theorems (Jackson and Bernstein inequalities) of the theory of approximation of functions. In this connection, the problem of precise estimates of the constants of such inequalities is very important [1,6,9,14].…”

Section: Introductionmentioning

confidence: 99%

Approximation of positive operators by analytic vectors

Dmytryshyn

2020

Carpathian Math. Publ.

View full text Add to dashboard Cite

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

show abstract

“…(d) By [7, Theorem 1(iv)] we have the completeness of E ν q (A) for q = 1, ∞. Then the space E ν q,p (A) is complete as an interpolation space according to (1).…”

mentioning

confidence: 97%

“…Finally, note that the applications of analytic vectors to approximation problems can be found in [8,10,11,14] and etc. As for exact constants in direct and inverse approximation theorems of the functions theory, see also [1,17].…”

mentioning

confidence: 99%

Approximation by interpolation spectral subspaces of operators with discrete spectrum

Dmytryshyn

2021

Mat. Stud.

View full text Add to dashboard Cite

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

show abstract

Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

Abstract: In the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces S p. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces S p .

Cited by 12 publications

References 14 publications

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Approximation of positive operators by analytic vectors

Approximation by interpolation spectral subspaces of operators with discrete spectrum

Contact Info

Product

Resources

About