## On a class of orthogonal series.(English)Zbl 0709.42016

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 V. A. Skvorcov’s paper [Mat. Sb., Nov. Ser. 4, No.1, 509-513 (1968)].
Reviewer: J.C.Georgiou

### MSC:

 42C15 General harmonic expansions, frames 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 26A39 Denjoy and Perron integrals, other special integrals

### Keywords:

Perron integration; orthogonal projection
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### References:

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