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Mat. Sb. (N.S.), 1972, Volume 88(130), Number 4(8), Pages 522–535 (Mi msb3194)  

This article is cited in 9 scientific papers (total in 9 papers)

On existence conditions for the Stieltjes integral

V. I. Matsaev, M. Z. Solomyak


Abstract: A modification of the definition of the Stieltjes integral $\int_0^1f dg$ is proposed, and it is shown that this integral exists if $g\in\operatorname{Lip}\alpha$, $f\in W_1^{1-\alpha}$, and $0<\alpha<1$ ($W_1^{1-\alpha}$ is the Sobolev–Slobodetskii class. It is shown that this integral defines a general form of a linear functional on $W_1^{1-\alpha}$ and on the class $\operatorname{Lip}_0\alpha$ of functions $g$ for which $g(x)-g(y)=o(|x-y|^\alpha)$. Applications to the integration of abstract functions and to the theory of double operator integrals are given.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1972, 17:4, 515–527

Bibliographic databases:

UDC: 517.394.1
MSC: Primary 26A39, 26A42; Secondary 26A16, 26A45, 28A40, 46E35
Received: 18.06.1971

Citation: V. I. Matsaev, M. Z. Solomyak, “On existence conditions for the Stieltjes integral”, Mat. Sb. (N.S.), 88(130):4(8) (1972), 522–535; Math. USSR-Sb., 17:4 (1972), 515–527

Citation in format AMSBIB
\Bibitem{MatSol72}
\by V.~I.~Matsaev, M.~Z.~Solomyak
\paper On existence conditions for the Stieltjes integral
\jour Mat. Sb. (N.S.)
\yr 1972
\vol 88(130)
\issue 4(8)
\pages 522--535
\mathnet{http://mi.mathnet.ru/msb3194}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=507422}
\zmath{https://zbmath.org/?q=an:0246.26005}
\transl
\jour Math. USSR-Sb.
\yr 1972
\vol 17
\issue 4
\pages 515--527
\crossref{https://doi.org/10.1070/SM1972v017n04ABEH001600}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Rybkin, “The spectral shift function, the characteristic function of a contraction, and a generalized integral”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 237–281  mathnet  crossref  mathscinet  zmath  isi
    2. A. M. Sedletskii, “On the summability and convergence of non-harmonic Fourier series”, Izv. Math., 64:3 (2000), 583–600  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Kats B., “The Inequalities for Polynomials and Integration Over Fractal Arcs”, Can. Math. Bul.-Bul. Can. Math., 44:1 (2001), 61–69  crossref  isi
    4. A. M. Sedletskii, “Bases of Exponentials in the Spaces $L^p(-\pi,\pi)$”, Math. Notes, 72:3 (2002), 383–397  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. F. S. Nasyrov, “Symmetric Integrals and Their Application in Financial Mathematics”, Proc. Steklov Inst. Math., 237 (2002), 256–269  mathnet  mathscinet  zmath
    6. A. M. Sedletskii, “Analytic Fourier Transforms and Exponential Approximations. II”, Journal of Mathematical Sciences, 130:6 (2005), 5083–5255  mathnet  crossref  mathscinet  zmath
    7. F. S. Nasyrov, “Symmetric integrals and stochastic analysis”, Theory Probab. Appl., 51:3 (2007), 486–503  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. S. S. Pukhov, “Bases of exponentials, sines and cosines in weighted spaces on a finite interval”, Izv. Math., 75:2 (2011), 413–443  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. G. V. Rozenblyum, “O matematicheskikh rabotakh Mikhaila Zakharovicha Solomyaka”, Algebra i analiz, 30:3 (2018), 3–29  mathnet  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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