On the Constant and Step in Jackson's Inequality for Best Approximations by Trigonometric Polynomials and by Haar Polynomials

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson's inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables.