The point‐wise convergence of general rational Fourier series

Let{α k}k = 1 ∞be a sequence included in a compact subset of the unit disk D ; we consider the rational Fourier series{φ k(e ix) }k = 0 ∞that are obtained by orthogonalization of the Blaschke product sequence { B 0 ( e ix ) = 1, B 1 ( e ix ), ⋯  , B n ( e ix ), ⋯  }, whereB n(e ix) = ∏ k = 1 ne ix − α k1 −α ¯ke ix.In order to study the point‐wise convergence of this rational Fourier series, let the partial sums of f  ∈  L 1 [ −  π , π ] be defined asS n( f ) ( x ) : = ∑ k = − n n⟨ f , φ k⟩φ k(e ix) ,whereφ − k(e ix) =φ k(e ix) ¯ for k ∈ N . In this paper, we will show that the conditions for the point‐wise convergence of f ( x ) −  S n ( f )( x ) is the same as that of the Fourier series. More precisely, one is the Dirichlet–Dini criterion, and the other is the Jordan test. Copyright © 2012 John Wiley & Sons, Ltd.