Singular integral operatörlerinin yaklaşım özellikleri

This thesis is a survey on some approximation properties of singular integral operatorsin the approximation theory. This thesis consists of four chapters. Firstly, a shorthistory of the studies on the approximation properties of singular integral and somebasic definitions and theorems were given. In the second chapter, the solution of theheat equation is the starting point of our study of Fourier series. Next a closer look atthe partial sums of Fourier series were given. Using the formula for the Fourier coefficients,the key observation was made such that these sums could be written as integralscalled singular integral of f with Dirichlet kernel or the convolution of f with Dirichletkernel, which led us to investigate the limiting properties of the convolutions as ntends to infinity. In this investigation, there found the kernels satisfying ApproximateIdentity whose convolutions with f tends to f . Unfortunately, Dirichlet kernel wasnot Approximate Identity. At this stage of the problem of convergence, various othermethods of summing the Fourier series of a function f could be considered that led toconvolutions of the function f with kernels satisfying Approximate Identity and in thethird chapter some important theorems about the convolutions of f with some kernelswere given. In the last chapter, the study of Assoc. Dr. Harun Karslı, which deals withthe theory of approximation of singular integral operators was further investigated.