On Several Problems Of Approximation Of Abstract Functions

In this thesis, we investigate several problems concerning approximation of abstract functions.In chapter 2, we define the abstract periodic three level bounded variation functions with range in a Banach space ?and study the property on approximating it by trigonometric polynomials with abstract coefficients from E. Finally the followingis proved: the order of approximation for x(t) ∈ V3 2π by the polynomials and thesufficient and necessary condition of x(t) ∈ V3 2π .In chapter 3, the property of approximation of general nonperiodic two level bounded variation function f(x) is discussed and that f(cos t) is a periodic twolevel bounded variation function is proved Secondly, the following is obtained: the order of approximation for f(x)∈ V2 [-1,1] by the polynomials and the sufficient andnecessary condition of f(x) ∈ V2 [-1,1 ] . Furthermore, We obtain more precise estimate of approximation for f(x) ∈ F2 [-1,1] by the algebra polynomials and the sufficientcondition of f(x) ∈ V2[-1, 1].In chapter 4, we study the property of several vector polynomials. In sector 1, we define the class of Chebyshev type vector polynomials and discuss an extremal property for Chebyshev type vector polynomials .It is proved that the Chebyshev type vector polynomials are also the least deviation functions and unique in the classwith some norms . In sector 2, we define the class PL and study the inequalities for derivatives of Lorentz type vector polynomials in B(p,q)-norm. A stronger estimate of higher derivatives for PL is obtained. In sector 3, we extend a theorem of Varma to B(p,q)-norm for some class of vector polynomials and obtain two similar results .