MEAN CES ` ARO-TYPE SUMMABILITY OF FOURIER-NEUMANN SERIES ´ OSCAR CIAURRI, KRZYSZTOF STEMPAK, AND JUAN L. VARONA Abstract. Let Jν be the Bessel function of order ν . For α> -1, the functions x -α-1 J α+2n+1 (x), n =0, 1, 2 ... , form an orthogonal system in L 2 (x 2α+1 dx), but the span of such functions is not dense in this space. For a function f , let S α k f denote the kth partial sum of the Fourier-Neumann series of f . In this paper we provide the minimal conditions on a real γ and 1 <p< ∞, for which the means R α n f = λ 0 S α 0 f +···+λnS α n f λ 0 +···+λn , λ k = 2(α +2k +2), are uniformly bounded in the spaces L p (x 2(α+γ)+1 dx). Clearly, the conver- gence R α n f → f holds only for functions from the closure of the linear span of the orthogonal system in these spaces. As a byproduct of the main result, we obtain a characterization of the closure of the span in terms of functions whose modified Hankel transforms of order α are supported on the interval [0, 1]. 1. Introduction and statement of results Let J ν stand for the Bessel function of the first kind of order ν . For α> −1, the formula  ∞ 0 J α+2n+1 (x)J α+2m+1 (x) dx x = δ nm 2(2n + α + 1) , n, m =0, 1, 2,..., provides an orthonormal system {j α n } ∞ n=0 in L 2 ((0, ∞),x 2α+1 dx)(L 2 (dμ α ) and, more generally, L p (dμ α ) from now on) given by j α n (x)=  2(2n + α + 1) J α+2n+1 (x)x −α−1 , n =0, 1, 2, .... For a function f , provided that the coefficients c α k (f )=  ∞ 0 f (y)j α k (y)y 2α+1 dy, k =0, 1, 2,..., exist, consider its formal expansion f ∼ ∑ ∞ k=0 c α k (f )j α k (x) and the partial sum operators S α n (f,x)= n  k=0 c α k (f )j α k (x), n =0, 1, 2, .... Series of the form ∑ n≥0 a n J α+n are usually called the Neumann series, hence we refer to ∑ ∞ k=0 c α k (f )j α k (x) as to the Fourier-Neumann series. For α and γ , α> −1, γ> −1 − α, let p 0 (α, γ ) = max  1, 4(α + γ + 1) 2α +3  , p 1 (α, γ )=  4(α+γ+1) 2α+1 , α> −1/2, ∞, −1 <α ≤−1/2. In the case γ = 0 we simply write p i (α) in place of p i (α, 0), i =1, 2. THIS PAPER HAS BEEN PUBLISHED IN: Studia Sci. Math. Hungar. 42 (2005), 413–430. 2000 Mathematics Subject Classification. Primary 42C10; Secondary 44A20. Key words and phrases. Riesz summation process, Ces` aro means, summability of Fourier- Neumann series, Hankel transform, Bessel functions. Research of the first and third authors supported by grant BFM2003-06335-C03-03 of the DGI. Research of the second author supported in part by KBN grant# 2 P03A 028 25. 1