Simultaneous and Converse Approximation Theorems in Weighted Orlicz Spaces

Akgün, Ramazan; İsrafilov, Daniyal M.

doi:10.36045/bbms/1267798496

Cited by 11 publications

(7 citation statements)

References 16 publications

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“…Using Lemma 1 we have (3.8) Ω β (f, t) p,ω ≤ Ω β (f − T n , t) p,ω + Ω β (T n , t) p,ω E n (f ) p,ω + t 2β T (2β) n p,ω . On the other hand using Theorem 1 of [4] and Theorem 5 we get (3.9) E n (f ) p,ω 1 n 2β E n f (2β) p,ω 1 n 2β Ω β f (2β) , 1/n p,ω t 2β f (2β) p,ω and the result follows from (3.8), (3.9), Theorem 1 of [3] and Theorem 5.…”

Section: Proof Of Theorems 3 Andmentioning

confidence: 81%

Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Akgün

2015

St. Petersburg Math. J.

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We obtain a characterization of modulus of smoothnes of fractional order in the Lebesgue spaces L p ω , 1 < p < ∞, with weights ω satisfying the Muckenhoupt's A p condition. Also, a realization result and equivalence between modulus of smoothness and the Peetre K-functional are proved in L p ω for 1 < p < ∞ and ω ∈ A p. Theorem 1 ([31]). Suppose that r > 0 and 1 ≤ p ≤ ∞. In this case (a) If f ∈ L p , then there exists a function ϕ ∈ Φ r such that ϕ (t) ≈ ω r (f, t) p holds for any t ∈ (0, ∞), where equivalence constants are depend only on r and p. (b) If ϕ ∈ Φ r , then there exist f ∈ L p and a positive real number t 0 such that ω r (f, δ) p ≈ ϕ (δ) holds for any δ ∈ (0, t 0 ], where the equivalence constants are depend only on r and p.

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Section: Proof Of Theorems 3 Andmentioning

confidence: 81%

Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Akgün

2015

St. Petersburg Math. J.

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“…In several particular cases there were some results of the Jackson type inequality: when γ ≡ 1 and p ∈ [1, ∞) (5) and 7in L p were considered in [11] and an equivalence of modulus of smoothness with Peetre's K -functional was proved. When γ ≡ 1 and p = 2 in L 2 Abilov and Abilova [1] obtained Theorem 4 thanks to the Parseval equality.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…The rest of the work is organized as follows. In Section 2 we give some preliminary properties of weights and the modulus of smoothness (5). In Section 3 we give the proof of Gadjieva's conjecture.…”

Section: Then There Exists a Positive Constant C Depending Only On Rmentioning

confidence: 99%

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Gadjieva's conjecture, K -functionals, and some applications in weighted Lebesgue spaces

Akgün¹

2018

Turk J Math

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We prove that Gadjieva's conjecture holds true as stated in her PhD thesis. The positive solution of this conjecture allows us to obtain improved versions of the Jackson-Stechkin type inequalities obtained in her thesis and some others. As an application, an equivalence of the modulus of smoothness with the realization functional is established. We obtain a characterization class for the modulus of smoothness.

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“…In weighted Lebesgue and Orlicz spaces with Muckenhoupt weights, these results were investigated in [24][25][26][27][28][29] . For more general doubling weights, some of these problems were investigated in 30 .…”

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confidence: 99%

Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Akgün

2012

Journal of Function Spaces and Applications

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Let and be, respectively, bounded and unbounded components of a plane curve satisfying Dini's smoothness condition. In addition to partial sum of Faber series of belonging to weighted Smirnov-Orlicz space (), we prove that interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .

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Simultaneous and Converse Approximation Theorems in Weighted Orlicz Spaces

Cited by 11 publications

References 16 publications

Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Gadjieva's conjecture, K -functionals, and some applications in weighted Lebesgue spaces

Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

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