A Sharp Jackson–Chernykh Type Inequality for Spline Approximations on the Line

An analog of the Jackson–Chernykh inequality for spline approximations in the space L 2 ( ) is established in this work. For r ∈ and σ > 0, we denote by A σ r ( f ) 2 the best approximation of a function f ∈ L 2 ( ) by the space of splines of degree r of minimal defect with knots , j ∈ , and by ω( f , δ) 2 the first-order modulus of continuity of f in L 2 ( ). The main result of our work is as follows. For any f ∈ L 2 ( ), where θ r = and ε r is the positive root of the equation The constant cannot be reduced on the whole class L 2 ( ) even by increasing the step of the modulus of continuity.