The goal of this paper is to establish the relations between general Bernstein and Nikol'skiî type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol'skiî type inequality is found. §1 IntroductionIn this paper, n is always a nonnegative integer and C is always an absolute positive constant. We also use C(a 1 , a 2 , a 3 , . . .) to denote a positive constant C depending only on a i (i = 1, 2, . . .). Let P n and T n be the sets of all algebraic and trigonometric polynomials of degree at most n with real coefficients, respectively. Let L p [a, b] be the space of real valued and p-integrable functions on [a, b] endowed with the norm [a,b] |f (x)|.As early as 1911, Bernstein in his doctoral dissertation[2] proved the inequality T n C[−π,π] ≤ n T n C [−π,π] , T n ∈ T n ,
In the paper titled as "Jackson-type inequality on the sphere" 2004 , Ditzian introduced a spherical nonconvolution operator O t,r , which played an important role in the proof of the wellknown Jackson inequality for spherical harmonics. In this paper, we give the lower bound of approximation by this operator. Namely, we prove that there are constants C 1 and C 2 such that C 1 ω 2r f, t p ≤ O t,r f − f p ≤ C 2 ω 2r f, t p for any pth Lebesgue integrable or continuous function f defined on the sphere, where ω 2r f, t p is the 2rth modulus of smoothness of f .
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