On a class of orthogonal series (Q5899972)

For \(n=0,1,..\). let \(D_ n\) be a finite set, \(\{0,1\}\subset D_ n\subset [0,1]\). Assume that \(D_ 0\subset D_ 1\subset..\). and that \(\cup D_ n\) is dense in [0,1]. Let \({\mathfrak D}\) be the sequence \(D_ 0,D_ 1,..\).. Let \({\mathcal D}_ n\) be the system of all components of \([0,1]\setminus D_ n\). Let \(V_ n\) be the system of all functions f on [0,1] such that f is constant on J for each \(J\in {\mathcal D}_ n\), \(f(0)=f(0+)\), \(f(1)=f(1-)\) and \(f(x)=(f(x+)+f(x-))\) for each \(x\in (0,1)\). Let \(T_ 0=V_ 0\) and for each \(n>0\) let \(T_ n\) be the orthogonal complement of \(T_{n-1}\) in \(V_ n\). For each \(x\in [0,1)\) [x\(\in (0,1]]\) let \(J_ n(x)[J^*_ n(x)]\) be the interval [a,b] for which (a,b)\(\in {\mathcal D}_ n\) and \(x\in [a,b)[x\in (a,b]]\); further set \(J_ n(1)=\{1\}\), \(J^*_ n(0)=\{0\}\) \((n=0,1,...).\) Using differentiation with respect to \({\mathfrak D}\) a \({\mathfrak D}\)- integration (and extension of the Perron integration on [0,1]) is defined. If T is a finite-dimensional space of piecewise constant functions on [0,1] and if f is a \({\mathfrak D}\)-integrable function, then the orthogonal projection of f to T is denoted by o.p. (f,T). - One of the results of the paper is the following assertion: Suppose that \(D_{n+1}\cap J\) has at most one point for each \(J\in {\mathcal D}_ n\) and that there is a number \(q>0\) such that \(d-c>q(b-a),\) whenever (a,b)\(\in {\mathcal D}_ n\), (c,d)\(\in {\mathcal D}_{n+1}\) and \((c,d)\subset (a,b)(n=0,1,...)\). Let \(f_ n\in T_ n\), \(s_ n=\sum^{n}_{k=0}f_ k\), \(\int_{J_ n(x)}s_ n\to 0\), \(\int_{J^*_ n(x)}s_ n\to 0\) (n\(\to \infty)\) for each \(x\in [0,1]\) and let the set \(\{x;\sup_ n| s_ n(x)| =\infty \}\) be countable. Then there is a \({\mathfrak D}\)- integrable function f such that \(\sum^{\infty}_{n=0}f_ n(x)=f(x)\) almost everywhere and that \(f_ n=o.p\). \((f,T_ n)\) for each n. A slight modification of this results leads to a generalization of Theorem 2 in \textit{V. A. Skvorcov}'s paper [Mat. Sb., Nov. Ser. 4, No.1, 509-513 (1968)].