Geodesic
Jackson type inequalities for differentiable functions in weighted Orlicz spaces
Algebra i analiz, Tome 34 (2022) no. 1, pp. 1-34

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In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt $A_{p}$ condition. To obtain a refined version of the Jackson type inequality, an extrapolation theorem, Marcinkiewicz multiplier theorem, and Littlewood–Paley type results are proved. As a consequence, refined inverse Marchaud type inequalities are obtained. By means of a realization result, an equivalence is found between the fractional order weighted modulus of smoothness and Peetre's classical weighted $K$-functional.
Keywords: Jackson inequality, moduli of smoothness, Muckenhoupt weight, trigonometric approximation. 
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R. Akgün. Jackson type inequalities for differentiable functions in weighted Orlicz spaces. Algebra i analiz, Tome 34 (2022) no. 1, pp. 1-34. http://geodesic.mathdoc.fr/item/AA_2022_34_1_a0/

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