Extension theorems

Theorems on the continuation (extension) of functions from one set to a larger set in such a way that the extended function satisfies certain definite properties. Problems on the analytic continuation of functions are, first of all, related to extension theorems.

An example of a theorem on the existence of a continuous extension of a continuous function is the Brouwer–Urysohn theorem: If $ E $ is a closed subset of a normal space $ X $ and $ f: E \rightarrow \mathbf R $ is a continuous real-valued bounded function, then there exists a continuous bounded function $ F: X \rightarrow \mathbf R $ such that $ F = f $ on $ E $. The Hahn–Banach theorem on the extension of linear functionals in vector spaces is an extension theorem.

In a Euclidean space extension theorems are mainly related to the following two problems: 1) the extension of functions with domain properly belonging to a space onto the whole space; and 2) the extension of functions from the boundary to the entire domain. In both cases it is required that the extended function has definite smoothness properties, i.e. belongs to an appropriate class of functions, depending on the properties of the function to be extended.

The problem of extending functions from a domain with a sufficiently smooth boundary to the whole space while preserving continuity of the partial derivatives was solved by M.R. Hestenes [3] and H. Whitney . If functions $ \phi _ {k} : \partial G \rightarrow \mathbf R $, $ k = 0 \dots m $, are given on the $ ( n - 1) $- dimensional boundary $ \partial G $ of a domain $ G $ in the $ n $- dimensional space $ \mathbf R ^ {n} $, then the problem of constructing a function $ u: G \rightarrow \mathbf R $ for which

$$ \tag{* } \frac{\partial ^ {k} u }{\partial n ^ {k} } = \ \phi _ {k} ,\ \ k = 0 \dots m,\ \ u \in C ^ \infty ( G), $$

where $ n $ is the normal to $ \partial G $, has been considered by E.E. Levi [5], G. Giraud [6],

and M. Gevrey [8] in case the smoothness of the $ \phi _ {k} $ and of $ \partial G $ is described in terms of continuity and membership of a Hölder space (in the presence of, possibly, certain singularities). The order of growth of the partial derivatives of orders $ k > m $ as the argument tends to the boundary $ \partial G $ of $ G $ has also been studied.

Both problems have been systematically studied by S.M. Nikol'skii and his students (cf. [9], [10]) in the cases of extension of functions in various $ L _ {p} $ metrics $ ( 1 \leq p \leq \infty ) $, in various variations and in various function spaces. Best characteristics of differentiability properties of functions that can be obtained from extending functions with given differentiability-difference properties have been found in terms of series of function spaces (cf. Imbedding theorems). Concerning the problem (*) one has found extensions that are optimal with respect to the order of growth of the derivatives of orders $ k > m $ when approaching the boundary of the manifold (cf. [11], ).

Often one substantiates methods for extending functions and systems of functions (*) from the boundary to the whole domain by integral representations. Usually, convenient methods for the extension of functions are linear. There are also other methods, e.g. based on expanding functions in series with subsequent extension of each term of the series. This method is, as a rule, non-linear. There are cases in which a linear method definitely does not exist, [13].

References

[1]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) MR1034865 MR0979016 MR0031025 Zbl 1175.01034 Zbl 45.0123.01
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
[3]	M.R. Hestenes, "Extension of the range of differentiable functions" Duke Math. J. , 8 (1941) pp. 183–192 MR3434
[4a]	H. Whitney, "Analytic extension of differentiable functions defined in closed sets" Trans. Amer. Math. Soc. , 36 (1934) pp. 63–89 MR1501735
[4b]	H. Whitney, "Differentiable functions defined in arbitrary subsets of Euclidean space" Trans. Amer. Math. Soc. , 40 (1936) pp. 309–317 MR1501875 Zbl 0015.01001 Zbl 62.0272.04 Zbl 62.0265.02
[5]	E.E. Levi, Mem. Soc. Itali XL , 16 (1909) pp. 3–112
[6]	G. Giraud, "Sur le problème de Dirichlet généralisé" Ann. Sci. Ecole Norm. Sup. , 46 (1929) pp. 131–245 MR1509295 Zbl 55.0285.02 Zbl 55.0284.03
[7a]	G. Giraud, "Sur certains problèmes non-linéaires de Neumann et sur certains problèmes non-linéaires mixtes" Ann. Sci. Ecole Norm. Sup. , 49 (1932) pp. 1–104 MR1509324 MR1509323 MR1509318 Zbl 0005.20502 Zbl 0004.39504 Zbl 58.0494.03
[7b]	G. Giraud, "Sur certains problèmes non-linéaires de Neumann et sur certains problèmes non-linéaires mixtes" Ann. Sci. Ecole Norm. Sup. , 49 (1932) pp. 245–309 MR1509324 MR1509323 MR1509318 Zbl 0005.20502 Zbl 0004.39504 Zbl 58.0494.03
[8]	M. Gevrey, "Les quasi-fonctions de Green et les systèmes d'équations aux dérivées partielles du type elliptique" Ann. Sci. Ecole Norm. Sup. , 52 (1935) pp. 39–108 MR1509347 Zbl 0011.40305 Zbl 61.0520.01
[9]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024
[10]	O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian) MR0519341 MR0521808 Zbl 0392.46022
[11]	L.D. Kudryavtsev, Trudy Mat. Inst. Steklov. , 55 (1956)
[12a]	S.V. Uspenskii, "Inclusion and extension theorems for a class of functions" Siberian Math. J. , 7 : 1 (1966) pp. 154–161 Sibirsk. Mat. Zh. , 7 : 1 (1966) pp. 192–199
[12b]	S.V. Uspenskii, "Embedding and extension theorems for one class of functions II" Siberian Math. J. , 7 : 2 (1966) pp. 333–342 Sibirsk. Mat. Zh. , 7 : 2 (1966) pp. 409–418
[13]	N.I. Burenkov, M.L. Gol'dman, "On extension of -functions" Proc. Steklov Inst. Math. , 150 (1967) pp. 33–54 Trudy Mat. Inst. Steklov. , 150 (1979) pp. 31–51 Zbl 0476.46030 Zbl 0417.46038 Zbl 0417.46037 Zbl 0357.46041 Zbl 0355.46014 Zbl 0351.46022

Comments

The Brouwer–Urysohn theorem is usually called the Tietze–Urysohn theorem or Tietze's extension theorem. It remains true if "bounded" is deleted twice.