On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\bar{\R}^m_+$

Details On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\bar{\R}^m_+$ On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\bar{\R}^m_+$

2011-12-21

arxiv.org/abs/1112.4950v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0}...\sum^\infty_{j_m=0} \ c_{j_1, j_2,..., j_m}\leqno(*)$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation. We introduce and investigate the regular convergence of the $m$-multiple integral $$\int^\infty_0 \int^\infty_0...\int^\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\leqno(**)$$ where $f: \bar{\R}^m_+ \to \C$ is a locally integrable function in Lebesgue's sense over the closed positive octant $\bar{\R}^m_+:= [0, \infty)^m$. Our main result is a generalized version of Fubini's theorem on successive integration formulated in Theorem 4.1 as follows. If $f\in L^1_{\loc} (\bar{\R}^m_+)$, the multiple integral (**) converges regularly, and $m=p+q$, where $m, p\in \N_+$, then the finite limit $$\lim_{v_{p+1},..., v_m \to \infty} \int^{v_1}_{u_1} \int^{v_2}_{u_2}...\int^{v_p}_{u_p} \int^{v_{p+1}}_0...\int^{v_m}_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m$$ $$=:J(u_1, v_1; u_2, v_2;...; u_p, v_p), \quad 0\le u_k\le v_k<\infty, \ k=1,2,..., p,$$ exists uniformly in each of its variables, and the finite limit $$\lim_{v_1, v_2,..., v_p\to \infty} J(0, v_1; 0, v_2;...; 0, v_p)=I$$ also exists, where $I$ is the limit of the multiple integral (**) in Pringsheim's sense.

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