Moduli of smoothness of higher orders related to the Fourier-Jacobi expansion and the approximation of functions by algebraic polynomials (Q1363286)

The author studies the modulus of smoothness of an arbitrary (fractional) order introduced with the generalized shift operator determined by the Fourier-Jacobi expansion, and shows its equivalence to a Peetre \(K\)-functional. Furthermore, for these moduli, he outlines the proof of the direct theorem of the theory of the best approximation of functions from \(L_{p,\alpha,\beta}[-1,1]\), where \(p\geq 1\) and \(\alpha\geq\beta\geq- 1/2\), by algebraic polynomials. An estimation for \(E_n(f)_{p,\alpha,\beta}\) is also given if \(f\in W^\ell_{p,\alpha,\beta}[-1,1]\).