×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Mařík, On a class of orthogonal series,Real Analysis Exchange,4 (1978–79), 53–57.
[2] M. A.Nyman,On a generalization of Haar series, Ph. D. Thesis, Mich. State University, 1972.
[3] S. Saks,Theory of the integral, Dover (New York, 1964). · Zbl 1196.28001
[4] V. A. Skvorcov, Calculation of the coefficients of an everywhere convergent Haar series,Math. USSR – Sbornik,4 (1968), 317–327. · doi:10.1070/SM1968v004n03ABEH002801
[5] V. A. Skvorcov, Differentiation with respect to nets and the Haar series,Math. Notes,4 (1968) 509–513.
[6] W. R. Wade, A uniqueness theorem for Haar and Walsh series,Trans. Amer. Math. Soc.,141 (1969), 187–194. · Zbl 0182.39604 · doi:10.1090/S0002-9947-1969-0243265-9
[7] W. R. Wade, Uniqueness theory for Cesàro summable Haar series,Duke Math. J.,38 (1971) 221–227. · Zbl 0221.42011 · doi:10.1215/S0012-7094-71-03828-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.