On Spectral Approximations of Unbounded Operators

Dmytryshyn, M.I.; Lopushansky, Oleh

doi:10.1007/s11785-019-00923-0

Cited by 6 publications

(6 citation statements)

References 15 publications

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“…and E(t, x; E q,p (A), X) = inf { ∥x − x 0 ∥ X : x 0 ∈ E q,p (A), |x 0 | Eq,p(A) ≤ t } for all x ∈ X. If q = p then E q,q (A) := E q (A) and we obtain the approximation spaces B s q,q,τ (A) =: B s q,τ (A), which were considered in [7,9]. Now let us define, for any x ∈ X and ν > 0,…”

Section: Estimates Of Spectral Approximation Errorsmentioning

confidence: 99%

“…The established inequalities fully characterize the subspace of elements from X in relation to rapidity of approximations. Note that exact estimates for approximation errors of spectral approximations for unbounded operators in Banach spaces, using the Besov-type quasi-norms and normalization factor N ′ θ,r = [rθ(1−θ)] 1/r for 1 ≤ r < ∞ and N ′ θ,∞ = 1, are given in [9]. N θ,r is also used in [5] to study the approximation problem by invariant subspaces of analytic vectors of positive operators in Banach spaces.…”

mentioning

confidence: 99%

“…The last section of this paper contains applications. Similarly to [9], we give new estimates of the spectral approximations errors for a regular elliptic operator in L q (Ω) over an open bounded set Ω ⊂ R n and for some self-adjoint ordinary differential boundary-value problems.…”

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confidence: 99%

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Approximation by interpolation spectral subspaces of operators with discrete spectrum

Dmytryshyn

2021

Mat. Stud.

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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.

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Section: Estimates Of Spectral Approximation Errorsmentioning

confidence: 99%

mentioning

confidence: 99%

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Approximation by interpolation spectral subspaces of operators with discrete spectrum

Dmytryshyn

2021

Mat. Stud.

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“…Analytic vectors of the unbounded linear operator on a Banach space first appear in [11]. It should be noted that the applications of analytic vectors to approximation problems can be found in [5][6][7][8] and ect. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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.

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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.

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Bernstein–Jackson Inequalities on Gaussian Hilbert Spaces

Lopushansky

2023

J Fourier Anal Appl

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Estimates of best approximations by exponential type analytic functions in Gaussian random variables with respect to the Malliavin derivative in the form of Bernstein–Jackson inequalities with exact constants are established. Formulas for constants are expressed through basic parameters of approximation spaces. The relationship between approximation Gaussian Hilbert spaces and classic Besov spaces are shown.

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On Spectral Approximations of Unbounded Operators

Cited by 6 publications

References 15 publications

Approximation by interpolation spectral subspaces of operators with discrete spectrum

Approximation by interpolation spectral subspaces of operators with discrete spectrum

Approximation of positive operators by analytic vectors

Bernstein–Jackson Inequalities on Gaussian Hilbert Spaces

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