Results Math (2018) 73:8 Online First c  2018 Springer International Publishing AG, part of Springer Nature https://doi.org/10.1007/s00025-018-0762-4 Results in Mathematics Approximation by Matrix Transforms in Weighted Lebesgue Spaces with Variable Exponent Daniyal M. Israfilov and Ahmet Testici Abstract. In this work the approximation properties of the matrix trans- forms of functions in the weighted variable exponent Lebesgue spaces are investigated. Mathematics Subject Classification. 41A10, 42A10. Keywords. Trigonometric approximation, matrix transforms, Muckenhoupt weights, modulus of smoothness, variable exponent Lebesgue spaces, Lipschitz classes. 1. Introduction Let T := [0, 2π] and let p (·): T → [0, ∞) be a Lebesgue measurable 2π periodic function. The variable exponent Lebesgue space L p(·) (T) is defined as the set of all Lebesgue measurable 2π periodic functions f such that ρ p(·) (f ) :=  2π 0 |f (x)| p(x) dx < ∞. During this work we suppose that the considered exponent functions p (·) satisfy the conditions 1 <p − := ess inf x∈T p (x) ≤ ess sup x∈T p (x) := p + < ∞, |p (x) − p (y)| ln (1/ |x − y|) ≤ c, x, y ∈ T, 0 < |x − y|≤ 1/2. (1) The class of these exponents we denote by P 0 (T). Equipped with the norm ‖f ‖ p(·) = inf  λ> 0: ρ p(·) (f /λ) ≤ 1  L p(·) (T) becomes a Banach space. 0123456789().: V,-vol