Oscillation and multilinear Stieltjes integral
Let \(Y\) and \(X_j\), \(y= 1,2,\dots,p,\) be normed spaces and \(A:X_1\times X_2\times \cdots X_p\to Y\) be a bounded multilinear transformation. Let \(g:[a,b]\to X_k\), and \(f_j:[a,b]\to X_j\), \(j = 1,\dots, k-1,k+1,\dots,p\), \(j\neq k\). Given a partition \(p =\{[t_{i-1},t_i]\}^n_{i=1}\) of \([a,b]\), and any \(p-1\) points from \([t_{i-1},t_i]\), namely \(s^j_i\), \(j=1,\dots,p\), \(j\neq k\), the Stieltjes sum \(S(p, \{s^j_i\})\) is defined as follows: NEWLINE\[NEWLINES(P,\{s^i_j\})= \sum^n_{i=1} A[f_1(s^1_i),\dots,f_{k-1}(s^{k-1}_i), g(t_i)-g(t_{i-1}), f_{k+1}(s^{k+1}_i),\dots,f_p(s^p_i)].NEWLINE\]NEWLINE For example, let \(M_{m,n}\) denote the normed space of all \(m\times n\) matrices. Let \(X_1= M_{2,3}, X_2 =M_{3,2}\), \(X_3=M_{2,4}\) and \(Y=M_{2,4}\). Suppose that \(A: X_1\times X_2\times X_3\to Y\) as ordinary matrix multiplication, i.e., \(A(Z_{23}, Z_{32},Z_{24})=Z_{23}\cdot Z_{32}\cdot Z_{24}\), where \(Z_{i,j}\) is a matrix in \(M_{i,j}\). Then NEWLINE\[NEWLINES(P,\{s^j_i\})=\sum^n_{i=1}f_1(s^1_i)\cdot [g(t_i)-g(t_{i-1})]\cdot f_3(s^3_i).NEWLINE\]NEWLINE Using the Stieltjes sums as above, multilinear Stieltjes integrals in the Riemann Moore-Pollard, Young-Moore-Pollard and Henstock-Kurzweil sense can be defined. Existence theorems for the first three cases have been proved by the author in ``Semi-variation and multilinear Stieltjes integrals'' [appeared in Glas. Mat., III. Ser. 32, No.~1, 17-28 (1997; Zbl 0893.28007)]. In this note, the proofs have been improved.
scientific article; zbMATH DE number 1658760
scientific article