The limit of Lebesgue constants for summation methods of the Fourier-Legendre series defined by a multiplier function

Let C[-1, 1] be the space of continuous functions f:[-1, 1] → ℝ with the uniform norm, let Pk be the Legendre polynomials such that Pk(1) = 1, and let J0 be the Bessel function of zero index. We consider sequences of linear operators (summation methods) U n:C[-1, 1] → C[-1, 1] defined by a multiplier function φ as follows: Un∫(y) = ∫-11 ∫(x)∑k=0∞φ(k/n)(k+1/2)P k(y)Pk(x)dx. The values ℒn, the norms of the operators Un, are called the Lebesgue constants of a summation method. The main result of this paper is the following statement. If a function φ is continuous on [0,+∞), ∑k=0∞φ 2(k/n)(k+1/2) < ∞ for each n ∈ ℕ, ∫0∞ φ2(x)x dx < ∞, Bφ(z) = z ∫0∞ φ(x)xJ0(zx) dx is the Fourier-Bessel transform of φ, and the function z q-1|Bφ(z)|q is summable on [0,+∞) for some q > 1, then limn→∞ ℒn = ∫0∞ |Bφ|. © 2002 Plenum Publishing Corporation.