The Limit of Lebesgue Constants for Summation Methods of the Fourier―Legendre Series Defined by a Multiplier Function

Abstract

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) _Un:C [-1,1]→ C[-1,1] defined by a multiplier function ϕ as follows:

$$U_n f(y) = \int_{ - 1}^1 {f(x)} \sum\limits_{k = 0}^\infty {\varphi ({k \mathord{\left/ {\vphantom {k n}} \right. \kern-\nulldelimiterspace} n})} (k + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})P_k (y)P_k (x)dx.$$

The values ${\mathfrak{L}}_n$, 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,+∞),

$$\sum\limits_{k = 0}^\infty {\varphi ^2 } ({k \mathord{\left/ {\vphantom {k {n)(k + {1 \mathord{\left/ {\vphantom {1 {2) < \infty {\text{ }}for{\text{ }}each}}} \right. \kern-\nulldelimiterspace} {2) < \infty {\text{ }}for{\text{ }}each}}}}} \right. \kern-\nulldelimiterspace} {n)(k + {1 \mathord{\left/ {\vphantom {1 {2) < \infty {\text{ }}for{\text{ }}each}}} \right. \kern-\nulldelimiterspace} {2) < \infty {\text{ }}for{\text{ }}each}}}}{\text{ }}n \in \mathbb{N},{\text{ }}\int_{\text{0}}^\infty {\varphi ^{\text{2}} } (x)xdx < \infty ,$$

$$B\varphi (z) = z\int_0^\infty {\varphi (x)xJ_0 } (zx)dx$$

is the Fourier―Bessel transform of ϕ, and the function $z^{q - 1} |B{\varphi (z)}|^q$ is summable on [\0,+∞) for some q>1, then

$${\mathop {\lim }\limits_{n \to \infty}} {\mathfrak{L}}_n = \int_0^\infty {|{\kern 1pt} B\varphi {\kern 1pt} |} .$$

Bibliography: 8 titles.