Uniform Approximations by Trigonometric Polynomials

Степанец, А. И.

doi:10.1515/9783110926033

Cited by 14 publications

(20 citation statements)

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“…In the present section we will define the general cosine-type approximation operators and the apparatus that is needed for estimating the order of approximation. The leading idea for definitions below appeared from the trigonometric approximation (see [14], [3], [12], [15]) and references cited there.…”

Section: Modulus Of Continuity Best Approximations General Cosine-tmentioning

confidence: 99%

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On a cosine operator function framework of approximation processes in Banach space

Kivinukk

Saksa

Zeltser

2019

Filomat

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We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

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Section: Modulus Of Continuity Best Approximations General Cosine-tmentioning

confidence: 99%

“…The next properties are adaptions of the well-known properties of the ordinary modulus of continuity (see, e.g. [3], [15], [17]).…”

Section: Modulus Of Continuity Best Approximations General Cosine-tmentioning

confidence: 99%

On a cosine operator function framework of approximation processes in Banach space

Kivinukk

Saksa

Zeltser

2019

Filomat

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“…де M -фiксований клас функцiй; ( ; ) -тригонометричнi полiноми, породженi методами пiдсумовування ряду Фур'є, а -простiр або , 1 ∞, називаємо (за О. I. Степанцем (Stepanets, 1981)), задачею Колмогорова -Нiкольського (K-H).…”

Section: задача фавараunclassified

Scientific heritage of Ukrainian mathematician V. K. Dzyadyk (to the centennial of the birth)

Zaderei¹,

Zaderei²,

Nefodova³

2018

MMTU

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18 лютого 2019 року виповнюється 100 рокiв вiд дня народження видатного українського математика члена-кореспондента НАН України Владислава Кириловича Дзядика, наукова спадщина якого мiстить значнi результати з теорiї наближення функцiй. Фундаментальнi працi вченого належать до конструктивної теорiї функцiй комплексної змiнної. У статтi розглянуто двi задачi, розв'язання яких принесли вченому свiтову славу. Перша-це задача Фавара про найкраще наближення функцiй з класiв при дробових , а друга-задача Колмогорова-Нiкольського про точнi верхнi гранi вiдхилень лiнiйних методiв пiдсумовування рядiв Фур'є на деяких класах. Висвiтлено також вагомий внесок В. К. Дзядика в розвиток школи з теорiї наближення функцiй, поєднання плiдної багаторiчної наукової творчої працi iз блискучою педагогiчною дiяльнiстю. Ключовi слова: задача Фавара; задача Колмогорова-Нiкольського; найкраще наближення; асимптотика верхнiх граней; константи Фавара; лiнiйний метод пiдсумовування рядiв Фур'є; суми Феєра; метод Рогозинського.

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“…If, in addition, for numbers r > 0, s > 0, r 1 ≥ r, and s 1 ≥ s, the conditions Ψ 1 ( k ) = k -r , Ψ 2 ( k ) = k -s , ψ 1 ( k ) = k -r 1 , and ψ 2 ( k ) = k -s 1 are satisfied, then the classes C m ∞ ψ coincide with the classes W r s r s 1 1 , , . In [10] (see also [3,11]), the problem of the approximation of the classes W r s r s …”

mentioning

confidence: 99%

“…Following [3] (see also [2,9]), we define rectangular linear means of Fourier series as follows: We associate a function f ∈ L ( T m ) that has the Fourier series (3) with the family of trigonometric polynomials…”

mentioning

confidence: 99%

Approximation of Classes of $${\bar \psi }$$ -Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series

2005

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Uniform Approximations by Trigonometric Polynomials

Cited by 14 publications

References 0 publications

On a cosine operator function framework of approximation processes in Banach space

On a cosine operator function framework of approximation processes in Banach space

Scientific heritage of Ukrainian mathematician V. K. Dzyadyk (to the centennial of the birth)

Approximation of Classes of $${\bar \psi }$$ -Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series

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