一些與凸函數相關的多重積分序列

Some Sequences of Multiple Integral Associated with Convex Functions

莊逸丞(Yi-Cheng Chuang)

指導教授 ： 陳功宇

淡江大學/理學院/數學學系碩士班/碩士(2017年)

https://doi.org/10.6846/TKU.2017.00742

摘要

令 f∶I⊆R→R 是有界的Lebesgue可積函數，g∶[a,b]→I 是一個連續函數，且對所有 n∈N，{q_i (n)∶1≤i≤n} 是一個正實數序列。
我們定義下列序列：
A_n (f,g;q)∶= 1/(b-a)^n ∫_([a,b]^n)f((q_1 (n)g(x_1 )+⋯+q_n (n)g(x_n ))/Q_n ) dx，
這裡 Q_n∶=∑_(i=1)^n〖q_i (n) 〗 ，dx=dx_1⋯dx_n 。
　　我們探討序列 A_n (f,g;q)的收斂及估計。

關鍵字

Jensen不等式 ； 凸函數 ； 多重積分序列

並列摘要

Let f∶I⊆R→R be a bounded Lebesgue integrable function, g∶[a,b]→I be continuous function, and for each n∈N, {q_i (n)∶1≤i≤n} is a sequence of positive real numbers.
We define the sequence:
A_n (f,g;q)∶= 1/(b-a)^n ∫_([a,b]^n)f((q_1 (n)g(x_1 )+⋯+q_n (n)g(x_n ))/Q_n ) dx，
where Q_n=∑_(i=1)^n〖q_i (n) 〗, dx=dx_1⋯dx_n.

We investigate the properties of convergence and estimation of the sequence A_n (f,g;q).

並列關鍵字

Jensen’s inequality ； Convex functions ； Sequences of multiple integral

參考文獻

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