The Research Of Fejer Trigonometric Interpolation

In this thesis, we discuss the fundamental study on the problem of Fejer-trigonometric interpolation process with equidistant points. We summarize variety forms of Fejer trigonometric interpolation process. The thesis consist three parts.In the first part, we show some basic knowledge and significant theorem of approximation theory, such as the best approximation theorem and the famous Chebyshev approximating polynomial, Cheybshev alternation points and Weierstrass approximation theorem. Then we discuss some approximation theory and approximation means.In the second part, we introduce the theory of Fejer trigonometric interpolation based on the theorem of Fejer sum. Firstly we presentation Fourier series and its sufficient conditions of uniform convergence. Then we discuss orthogonality of trigonometric function and the relation between Fourier series and trigonometric polynomial. Depending on partial summation of Fourier series, we give theorem of Fejer and the integral expression of Fejer sums. After that we recommend the better convergence of Fejer summation sequences. Finally we consider various construction of Fejer trigonometric interpolation polynomial, which include some discussion about the summability rule of Fejer summation sequences,and we also generalize the Fejer trigonometric interpolation process. Any work above is meaningful.In the third part, we show some new progress of Fejer trigonometric interpolation theory at present. We study Hermite-Fejer trigonometric interpolation and Lagrange trigonometric interpolation, moreover we show the transform between Hermite-Fejer trigonometric interpolation and Lagrange trigonometric interpolation. At last we investigate the uniform convergence of Fejer trigonometric interpolation, and explain how to apply Fejer trigonometric interpolation to picture processing.