Throughout this paper, all functions and vector spaces we shall consider are complex, m and n are fixed integers ≥1, E ≡ ℝn is the Euclidean n-space of all n-tuples of real numbers, P≡Pm-1 is the (complex) vector space of dimension
$$(\mathop n\limits^{m + n - 1} )$$
of all polynomials in n variables of (total) degree ≤m-1, |.| is the (Sobo-lev-like) seminorm (of kernel P) generated by the rotation invariant semi-inner product (on various suitable spaces of distributions to be specified later) where
$$(v,w): = \sum\limits_{{i_1},...,{i_m} = 1}^n {\int_{{\mathbb{R}^n}} {{\partial _i}.{i_m}v( \times ){\partial _{{i_1}}}} ...{i_m}\mathop w\limits^ - (x)dx}$$
where \({\partial _{{i_1}}}...{i_m}: = {\partial ^m}/\partial {x_{{i_1}}}...\partial {x_{{i_m}}}\) is to be interpreted in the distributional sense. All integrals will be taken with respect to the Lebesgue measure on E (this is quite natural in view of the importance of translations of E). As usual, V denotes the vector space of test functions in E (i.e., infinitely differentiable functions with compact support in E), provided with the canonical Schwartz topology, while V (i.e., the dual of V) is the vector space of distributions in E.


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© Springer Basel AG 1979

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  • Jean Meinguet

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