Approximation of functions of higher smoothness by Fourier sums

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of 2π-periodic functions, representable in the form of convolutions of functions ϕ, which belong to unit balls of spaces L p , with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.

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“…where |γ n | < 2 1 + 1 αrn r−1 e −αrn r−1 . Later Telyakovskii [6] established the asymptotic equality…”

mentioning

confidence: 99%

Uniform Approximations by Fourier Sums in Classes of Generalized Poisson Integrals

Serdyuk

Stepanyuk

2018

Anal Math

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“…Equalities (158')and (160) with a remainder in a slightly different form were proved earlier by Telyakovskii [23]. tions from L satisfying the condition H q)(" + t) -q~(.)…”

Section: T--)~ ~Lti(t)mentioning

confidence: 93%

Rate of convergence of Fourier series on the classes of $$\bar \Psi $$ -integrals-integrals

Степанец¹

1997

Ukr Math J

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We introduce the notion of ~-integrals of 2~:-periodic summable functions f, f ~ L, on the basis of which the space L is decomposed into subsets (classes) /.,~. We obtain integral representations of deviations of the trigonometric polynomials Un (f; x ; A) generated by a given A-method for summing the Fourier series of functions f ~ L vT. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets L ~ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets L ~, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.

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“…In the case p = ∞, Theorem 4 with the remainder written in a different (more exact) form was proved by Telyakovskii in [14]. The statement of Theorem 4 for p = 2 should also be regarded as known (see [2, pp.…”

Section: Approximation By Fourier Sums On Classes Of Entire Functionsmentioning

confidence: 97%

“…, and, hence, taking into account inequalities (56) and the obvious relations To prove relation (14) it suffices to note that 2 2…”

Section: Lemma 1 Let 1 ≤ S ≤ ∞ and Let 2π-periodic Functions G ( T )mentioning

confidence: 99%

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Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

Serdyuk

2005

Ukr Math J

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Approximation of functions of higher smoothness by Fourier sums

Cited by 20 publications

References 1 publication

Uniform Approximations by Fourier Sums in Classes of Generalized Poisson Integrals

Uniform Approximations by Fourier Sums in Classes of Generalized Poisson Integrals

Rate of convergence of Fourier series on the classes of $$\bar \Psi $$ -integrals-integrals

Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

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