Abstract
The validity of Clarkson’s inequalities for periodic functions in the Sobolev space normed without the use of pseudodifferential operators is proved. The norm of a function is defined by using integrals over a fundamental domain of the function and its generalized partial derivatives of all intermediate orders. It is preliminarily shown that Clarkson’s inequalities hold for periodic functions integrable to some power p over a cube of unit measure with identified opposite faces. The work is motivated by the necessity of developing foundations for the functional-analytic approach to evaluating approximation methods.
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References
S. L. Sobolev, Introduction to the Theory of Cubature Formulas (Nauka, Moscow, 1974) [in Russian].
J. A. Clarkson, “Uniformly convex spaces,” Trans. Am. Math. Soc. 40, 396–414 (1936). doi 10.2307/1989630
S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Nauka, Moscow, 1988) [in Russian].
O. Hanner, “On the uniformconvexity of Lp and lp,” Ark. Mat. 3, 239–244 (1956). doi 10.1007/BF02589410
P. Enflo, “Banach spaces which can be given an equivalent uniformly convex norm,” Isr. J. Math. 13, 281–288 (1972). doi 10.1007/BF02762802
R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renormings in Banach Spaces (Longman, Harlow, 1993).
V. R. Portnov, “Sobolev projection operators for seminorms with infinite-dimensional kernels,” Tr. MIAN SSSR 140, 252–263 (1976).
V. R. Portnov, “On some integral inequations,” in Embedding Theorems and Their Applications (Nauka, Moscow, 1970), pp. 195–203 [in Russian].
V. R. Portnov, “On a certain projection operator of Sobolev type,” Dokl. AN SSSR 189, 258–260 (1969).
M. S. Agranovich, Sobolev Spaces, their Generalizations and Elliptic Problems in Domains with Smooth and Lipschitz Boundary (MTsNMO, Msocow, 2013) [in Russian].
V. G. Maz’ya, Sobolev Spaces (Leningr. Gos. Univ., Leningrad, 1985) [in Russian].
Ts. B. Shoinzhurov, “The theory of cubature formulas in function spaces with the norm depending on the function and its derivatives,” Doctoral (Phys.Math.) Dissertation (Ulan-Ude, 1977).
Ts. B. Shoinzhurov, Estimation of Norm of Cubature Formula Error Functional in Various Functional Spaces (Buryat. Nauch. Tsentr SO RAN, Ulan-Ude, 2005) [in Russian].
Ts. B. Shoinzhurov, Cubature Formulae in Sobolev Space W p m (Vost. Sib. Gos. Tekh. Univ., Ulan-Ude, 2002) [in Russian].
I. V. Korytov, “Function representing error functional of a cubature formula in Sobolev space,” Izv. Tomsk. Politekh. Univ. 323 (2), 21–25 (2013).
I. V. Korytov, “The extreme function of a linear functional at the weighted Sobolev space,” Vestn. Tom. Univ., Mat. Mekh., No. 2, 5–15 (2011).
I. V. Korytov, “Representation of error functional of cubature formula at weighted Sobolev space,” Vychisl. Tekhnol. 11, 59–66 (2006).
S. L. Sobolev and V. L. Vaskevich, Cubature Formulas (Inst. Matem., Novosibirsk, 1996) [in Russian].
V. L. Vaskevich, “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures,” Sib. Math. J. 53, 996–1010 (2012). doi 10.1134/S0037446612060043
A. Mbarki, A. Ouahab, and I. E. Hadi, “Convexity and fixed point properties in spaces of Bochner integrals nuclear-valued functions,” Appl.Math. Sci. 8 (84), 4179–4186 (2014).
H. Mizuguchi and K. S. Saito, “A note on Clarkson’s inequality in the real case,” J. Math. Inequalities 4, 29–132 (2010). doi 10.7153/jmi-04-13
T. Formisano and E. Kissin, “Clarkson–McCarthy inequalities for lp-spaces of operators in Schatten ideals,” Integr. Equations Oper. Theory 79, 151–173 (2014). doi 10.1007/s00020-014-2145-x
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Korytov, I.V. Clarkson’s inequalities for periodic Sobolev space. Lobachevskii J Math 38, 1146–1155 (2017). https://doi.org/10.1134/S1995080217060178
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DOI: https://doi.org/10.1134/S1995080217060178