Generalized Functions Of Bounded Variation Class L_p Continuous Mode Is Estimated That

The main aim of this dissertation is to investigate estimates of Lp modulus of continuity w(f; t)_p (1 ≤ p < ∞) of the class of functions of Λφ-bounded variation and its special cases.As a generalization of the well-known function of bounded variation, the definition of functions of Λφ-bounded variation combines the notions of function of λ-bounded variation and φ—bounded variation, and many important classes of functions are special cases of the class of functions of Λφ-bounded variation. Among them are classes of functions of A—bounded variation, φ—bounded variation, β—bounded variation and harmonic bounded variation etc.Characterization of approximative properties of some classes of functions, embedding relations between certain classes of functions and the class H_p~ω or the Lipschitz class H_p~α, and, summability and convergence of trigonometric series depend on estimates of L_p modulus of continuity of the corresponding classes of functions in many research topics in Approximation Theory. It is of significant importance to estimate L_p modulus of continuity of some important classes of functions.For the case of that φ is convex, we obtain some estimates of L_p modulus of continuity of class of functions of Λ—bounded variation and φ—bounded variation and its some special cases. Our estimates are sharp in the sense of order in most cases.As applications of our estimates, we characterize some sufficient and necessary conditions of embedding relations of Λφ BV (?) H_p~ω (1 ≤ p < ∞)and φBV C H_p~ω (1 ≤ p < ∞). Estimates of orders of Fourier coefficients of functions of Λ—bounded variation and φ—bounded variation are also obtained. Our results include the corresponding results of the class ΛφBV as a special case and also give the coresponding estiments of that of the class φBV. Our results are expected to have applications in trigonometric series and other research fields.