Clarkson's inequalities for periodic Sobolev space

I. V. Korytov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 3, pp. 336-349

Résumé

The paper is devoted to developing the proof of Clarkson's inequalities for periodic functions belonging to the Sobolev space. The norm of the space has not been considered earlier. The importance of the discussed issue rests with the need to develop fundamentals in research of error estimation using functional analysis techniques. Thus, the error of approximation is represented via a linear functional over the Banach space. The approach allows searching for new criteria of approximation quality and ways to optimize the numerical method. Parameters that are responsible for technique quality need to be previously investigated in respect of extreme values. Therefore, fundamental features, such as uniform convexity, should be proved for being further used in extremum problems solving. The aim of the study is to prove the uniform convexity for the Sobolev space of periodic functions normed without pseudodifferential operators. The norm includes integrals over the fundamental cube. The integrands are the absolute values of derivatives of all orders raised to the $p$-th power. The exponent p generates the non-Hilbert space. The methods used in the study include application of inverse Minkowski inequalities stated either for sums or for integrals to periodic functions from the Sobolev space. Furthermore, functional analysis concepts and techniques are used. As a result, Clarkson's inequalities are proved for periodic functions from the Sobolev space. The obtained results are important for solving extremum problems. The problems require using only uniformly convex functional spaces, for example, the problem of error estimation of numerical integration of functions from the Sobolev space with the above norm.

Détail

Keywords: uniform convexity, Banach space, Sobolev space, non-Hilbert space, periodic function space, inverse Minkowski inequality, unit cube, Clarkson's inequalities.

Bibliographie

[1] Sobolev S. L., Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974, 808 pp. (In Russian) | MR

[2] Clarkson J. A., “Uniformly convex spaces”, Trans. Am. Math. Soc., 40:3 (1936), 396–414 | DOI | MR | Zbl

[3] Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Nauka, Moscow, 1988, 336 pp. (In Russian)

[4] Hanner O., “On the uniform convexity of $L_p$ and $l_p$”, Ark. Mat., 3:3 (1956), 239–244 | DOI | MR | Zbl

[5] Enflo P., “Banach spaces which can be given an equivalent uniformly convex norm”, Isr. J. Math., 13:3 (1972), 281–288 | DOI | MR

[6] Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Longman, Harlow, 1993, 376 pp. | MR | Zbl

[7] Portnov V. R., “Sobolev projection operators for seminorms with infinite-dimensional kernels”, Proc. Steklov Inst. Math., 140 (1979), 277–288 | MR | Zbl

[8] Portnov V. R., “On some integral inequalities”, Embedding Theorems and Their Applications, Moscow, 1970, 195–203 (In Russian) | MR

[9] Portnov V. R., “On a certain projection operator of Sobolev type”, Dokl. Akad. Nauk SSSR, 189:2 (1969), 258–260 (In Russian) | Zbl

[10] Agranovich M. S., Sobolev Spaces, Their Generalizations and Elliptic Problems in Domains with Smooth and Lipschitz Boundary, Izd. MTsNMO, Moscow, 2013, 379 pp. (In Russian)

[11] Maz'ya V. G., Sobolev Spaces, Izd. Leningr. Univ., Leningrad, 1985, 416 pp. (In Russian)

[12] Shoynzhurov Ts. B., The theory of cubature formulas in function spaces with the norm depending on the function and its derivatives, Doct. Phys.-Math. Sci. Diss., Ulan Ude, 1977, 235 pp. (In Russian)

[13] Shoynzhurov Ts. B., Estimation of Norm of Cubature Formula Error Functional in Various Functional Spaces, Izd. Buryat. Nauchn. Tsentr Sib. Otd. Ross. Akad. Nauk, Ulan Ude, 2005, 247 pp. (In Russian)

[14] Shoynzhurov Ts. B., Cubature Formulas in Sobolev Space $W_p^m$, Izd. VSGTU, Ulan Ude, 2002, 201 pp. (In Russian)

[15] Korytov I. V., “Function representing error functional of a cubature formula in Sobolev space”, Byull. Tomsk. Polytekh. Univ., 323:2 (2013), 21–25 (In Russian)

[16] Korytov I. V., “The extreme function of a linear functional at the weighted Sobolev space”, Vestn. Tomsk. Gos. Univ., Mat. Mekh., 2011, no. 2(14), 5–15 (In Russian)

[17] Korytov I. V., “Representation of error functional of cubature formula at weighted Sobolev space”, Vychisl. Tekhnol., 11, Spec. issue (2006), 59–66 (In Russian) | Zbl

[18] Sobolev S. L., Vaskevich V. L., Cubature Formulas, Izd. Inst. Mat., Novosibirsk, 1996, 483 pp. (In Russian)

[19] Vaskevich V. L., “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures”, Sib. Math. J., 53:6 (2012), 996–1010 | DOI | MR | Zbl

[20] Mbarki A., Ouahab A., Hadi I. E., “Convexity and fixed point properties in spaces of Bochner integrals nuclear-valued functions”, Appl. Math. Sci., 8:84 (2014), 4179–4186 | DOI

[21] Mizuguchi H., Saito K. S., “A note on Clarkson's inequality in the real case”, J. Math. Inequalities, 4:1 (2010), 29–132 | MR

[22] Formisano T., Kissin E., “Clarkson–McCarthy inequalities for $l_p$-spaces of operators in Schatten ideals”, Integr. Equations Oper. Theory, 79:2 (2014), 151–173 | DOI | MR | Zbl