Convergence and integrability of rational and double rational trigonometric series with coefficients of bounded variation of higher order

Abstract

Abstract We prove that a rational trigonometric series with its coefficients c ⁢ ( n ) = o ⁢ ( 1 ) {c(n)=o(1)} satisfying the condition ∑ n ∈ ℤ | Δ m ⁢ c ⁢ ( n ) | < ∞ {\sum_{n\in\mathbb{Z}}|\Delta^{m}c(n)|<\infty} , m ∈ ℕ {m\in\mathbb{N}} , converges pointwise to some f ⁢ ( x ) {f(x)} for every x ∈ ( 0 , 2 ⁢ π ) {x\in(0,2\pi)} and also converges in L p ⁢ [ 0 , 2 ⁢ π ) {L^{p}[0,2\pi)} -metric to f for 0 < p < 1 m {0<p<\frac{1}{m}} . This result is further extended to a double rational trigonometric series.