Geodesic
Several properties of generalized multivariate integrals
Sbornik. Mathematics, Tome 198 (2007) no. 7, pp. 967-991 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Several properties of generalized multivariate integrals are considered. In the two-dimensional case the consistency of the regular Perron integral is proved, as well as the consistency of a generalized integral solving the problem of the recovery of the coefficients of double Haar series in a certain class. Several generalizations of Skvortsov's well-known theorem are obtained as consequences, for instance, the following result: if a double Haar series converges for some $\rho\in(0,1/2]$ $\rho$-regularly everywhere in the unit square to a finite function that is Perron-integrable in the $\rho$-regular sense, then the series in question is the Fourier–Perron series of its sum. Bibliography: 20 titles. 
@article{SM_2007_198_7_a3,
     author = {M. G. Plotnikov},
     title = {Several properties of generalized multivariate integrals},
     journal = {Sbornik. Mathematics},
     pages = {967--991},
     year = {2007},
     volume = {198},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_7_a3/}
}
TY  - JOUR
AU  - M. G. Plotnikov
TI  - Several properties of generalized multivariate integrals
JO  - Sbornik. Mathematics
PY  - 2007
SP  - 967
EP  - 991
VL  - 198
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2007_198_7_a3/
LA  - en
ID  - SM_2007_198_7_a3
ER  - 
%0 Journal Article
%A M. G. Plotnikov
%T Several properties of generalized multivariate integrals
%J Sbornik. Mathematics
%D 2007
%P 967-991
%V 198
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2007_198_7_a3/
%G en
%F SM_2007_198_7_a3
M. G. Plotnikov. Several properties of generalized multivariate integrals. Sbornik. Mathematics, Tome 198 (2007) no. 7, pp. 967-991. http://geodesic.mathdoc.fr/item/SM_2007_198_7_a3/

[1] P. du Bois-Reymond, “Beweis, dass die Coefficienten der trigonometrischen Reihe $f(x)=\varSigma(a_p\cos px+b_p\sin.px)$ die Werthe $a_0=\frac1{2\pi}\int_\pi^{+\pi}d\alpha f(\alpha)$, $a_p=\frac1\pi\int_\pi^{+\pi}d\alpha f(\alpha)\cos.p\alpha$, $b_p=\frac1\pi\int_\pi^{+\pi}d\alpha f(\alpha)\sin.p\alpha$, haben, jedesmal, wenn diese Integrale endlich und bestimmt sind”, Munch. Abh., XII:1 (1876), 117–166 | Zbl

[2] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 ; N. K. Bary, A treatise on trigonometric series, vols. I, II, Pergamon Press, Oxford–London–New York–Paris–Frankfurt, 1964 | MR | MR | Zbl

[3] V. A. Skvortsov, “O mnozhestvakh edinstvennosti dlya mnogomernykh ryadov Khaara”, Matem. zametki, 14:6 (1973), 789–798 | MR | Zbl

[4] V. A. Skvortsov, “Ob odnoi teoreme edinstvennosti dlya mnogomernogo ryada Khaara”, Izv. AN ArmSSR. Matem., 23:3 (1988), 293–296 | MR | Zbl

[5] V. A. Skvortsov, A. A. Talalyan, “Nekotorye voprosy edinstvennosti kratnykh ryadov po sisteme Khaara i trigonometricheskoi sisteme”, Matem. zametki, 46:2 (1989), 104–113 | MR | Zbl

[6] M. G. Plotnikov, “Voprosy edinstvennosti dlya kratnykh ryadov Khaara”, Matem. sb., 196:2 (2005), 97–116 | MR | Zbl

[7] P. L. Ulyanov, “O ryadakh po sisteme Khaara”, Matem. sb., 63(105):3 (1964), 356–391 | MR | Zbl

[8] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Izd. 2, Izd-vo AFTs, M., 1999 ; B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989 | MR | Zbl | MR | Zbl

[9] K. M. Ostaszewski, “Henstock integration in the plane”, Mem. Amer. Math. Soc., 63:353 (1986) | MR | Zbl

[10] L. V. Linkov, “Ekvivalentnost razlichnykh opredelenii dvoichnogo regulyarnogo integrala na ploskosti”, Vestn. MGU. Ser. 1. Matem., mekh., 2000, no. 5, 50–53 | MR | Zbl

[11] L. Di Piazza, “Variational measures in the theory of the integration in $\mathbb R^m$”, Czechoslovak Math. J., 51:1 (2001), 95–110 | DOI | MR | Zbl

[12] B. Bongiorno, L. Di Piazza, V. Skvortsov, “A new full descriptive characterization of Denjoy–Perron integral”, Real Anal. Exchange, 21:2 (1995), 656–663 | MR | Zbl

[13] Z. Buczolich, W. F. Pfeffer, “Variations of additive functions”, Czechoslovak Math. J., 47:3 (1997), 525–555 | DOI | MR | Zbl

[14] W. F. Pfeffer, “A descriptive definition of a variational integral and applications”, Indiana Univ. Math. J., 40:1 (1991), 259–270 | DOI | MR | Zbl

[15] W. F. Pfeffer, “Comparing variations of charges”, Indiana Univ. Math. J., 45:3 (1996), 643–654 | DOI | MR | Zbl

[16] C.-A. Faure, J. Mawhin, “The Hake's property for some integrals over multidimensional intervals”, Real Anal. Exchange, 20:2 (1995), 622–630 | MR | Zbl

[17] V. A. Skvortsov, “Nekotoroe obobschenie integrala Perrona”, Vestn. MGU. Ser. 1. Matem., mekh., 24:4 (1969), 48–51 | MR | Zbl

[18] M. G. Plotnikov, “O vosstanovlenii koeffitsientov dvumernykh ryadov Khaara”, Izv. vuzov. Ser. matem., 2005, no. 2, 45–53 | MR | Zbl

[19] M. G. Plotnikov, “O edinstvennosti vsyudu skhodyaschikhsya kratnykh ryadov Khaara”, Vestn. MGU. Ser. 1. Matem., mekh., 2001, no. 1, 23–28 | MR | Zbl

[20] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Lan, SPb., 1999 ; I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publ. Co., New York, 1955 | MR | Zbl | MR | Zbl