Let T_n(x) be an arbitrary trigonometric polynomial T_n(x)=1/2a_0+{sum_(k=1)^n[a_kcos(kx)+b_ksin(kx)]} (1) with real coefficients, let f be a function that is integrable over the interval [-pi,pi], and let the rth derivative of f be bounded in [-1,1]. Then there exists a polynomial T_n(x) for which |f(x)-T_n(x)|<=(K_r)/((n+1)^r), (2) for all x in [-pi,pi], where K_r is the smallest constant possible, known as the rth Favard constant. K_r can be given explicitly by the sum ...