Approximation Theory Of Classic And Cutting-edge Issues

The present thesis consists of three parts:Part â… . Approximation by rational functions with polynomials of real coefficients or positive coefficients as the denomintors and with prescribed numerator degree. We study some important topics such as the L_p-approximation, pointwise estimate and copositive approximation. The results obtained and the methods are interesting and have their values in theories and applications. We also raise some problems for further study. We believe that it needs some new ideas and methods to further development.Part â…¡. Turan type inequalities. First, we establish Turan type inequalities for rational functions with prescribed poles, and essentially improve the results of [Min], thus give a positive answer to a problem of Min. Secondly, we establish Turan type inequalities with doubling weights and A* weights. As we know, these are weights having inner singularities. Our method is also an simplification of the classic cases.Part â…¢. Improvement of some classical results in Fourier analysis. First, based on the condition of GBV raised by Zhou and Le, we suggest a new kind of condition-XBV condition, which transforms the monotonicity from "one-sided" to "two-sided". We study the relations among MS, QMDS, RBVS, AMDS, GBVS and NBVS. Finally. we improve some classic results in Fourier analysis such as uniform convergence, L~1 convergence, L~p-integrability, strong approximation of continuous functions et al., by applying this new condition.