Approximation of the functions by means of Fourier seriesin some function spaces

Fourier series in the Orlicz spaces. Obtained results are given in terms Ef best approximation and modulus of smoothness. In addition, the relationship between the Zygmund means of the Fourier series of functions in the space of continuous functions and the best approximation of the function in the Lebesgue space is investigated. Also, the modulus of smoothness of the function in the space of continuous functions is estimated from above by the approximation of the Zygmund means of the Fourier series of the function. In this study the approximation of the functions by linear means of Fourier series in weighted Orlicz spaces was investigated. This result was applied to the approximation of the functions by linear means of Faber series in weighted Smirnov-Orlicz classes defined on simply connected domain of the complex plane. We investigate the simultaneous approximation properties De la Vallée- Poussin means in Musielak-Orlicz spaces in terms of the modulus of smoothness. In terms of the modulus of smoothness the direct theorem of simıltaneous approximation is proved. Also in Musielak-Orlicz spaces the modulus of smoothness are estimated from below and above in terms n th partial sums and De la Vallée-Poussin means