By N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)

ISBN-10: 1475746660

ISBN-13: 9781475746662

ISBN-10: 1475746687

ISBN-13: 9781475746686

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N). о is selected so that hk" 1-\1( lк .. ~ .... ,n), we oЬtain n .!. .!.. k (к=-\, ... , n). \m+i, ... mн = ~~ ... rt. the inequality From this, with the aid of inequality (9'), we oЬtain (20) For entire functions of finite degree, for G=Е " , and for trigonometric polynomials, taken over а period, inequalities (19) and (20), with the indicated values of the constants С, were first oЬtained Ьу S. М. Nikol'skii [6] Ьу another method. ~, ... , Ьу а number of other authors. ·, G' i v; G) PtL·,G'; v·, G).

L- \ . Let the solution u. ) . ;. are functions of the corresponding classes P(t·,G'; v; 6-) . of the form (1 О) (for certain particular types of domains G, the functions simple algebraic polynomials). , Ье an arbltrary function of the form given in Eq. 4 , ... n ). L>-rlщ)-. ,. The possibllity of oьtaining an estimate for Et,t. ,,w;(G), (31) is based on the inequality (31) and theorems giving an estimate for \U. i. "wt. G)' An estimate for Ru. ~ ll,i (G) may Ье oьtained Ьу using а theorem of I.

LQ')' and this, the limiting form of Eq. (52) will Ье \ ( 4'"' \j1 and r in L, lQ'), converges to ct> uniformly in от·. Thanks to ~~ • У~ U ' Щ. ) U:ч т U;. \1""< - cr 1q> dx. т, whence, in view of the sufficient arЬitrariness of ф and the fact that u belongs to w~·~·а~ l ) ' it follows that t1 satisfies Eqs. (3). Thus, we have proved the following theorem: Theorem 1. )• and U"'IUw:·\Qт). The norm of the solution 'U, namely, nunw'·'lQT•)• is also determined Ьу these same quantities. " and the boundaries 51 of the domain nt in Е~ must have uniformly bounded "norms" in е for all t Е: [ОТ) and must satisfy conditions 1) and 2), given on page 42.

### Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory by N. M. Ivochkina, A. P. Oskolkov (auth.), O. A. Ladyzhenskaya (eds.)

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