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A characterization of potential spaces. (English) Zbl 0577.46035
Functions in the Bessel potential spaces $$L^ p_ a$$ are characterized in terms of their approximability by polynomials. More precisely, if f is locally in $$L^ 1$$, define for $$t>0$$, $$x\in R^ n$$ $E_ kf(x,t)=Ef(x,t)=\sup_{x\in Q,| Q| =t^ n}(\inf_{P\in {\mathbb{P}}^ k}\int_{Q}\quad | f-P| dz/| Q|)$ with $${\mathbb{P}}^ k$$ the polynomials of degree $$\leq k$$. Then we prove (theorem 2) that for $$a>0$$, $$k=[a]$$ and $$1<p<\infty$$, $$f\in L^ p_ a$$ iff $$f\in L^ p$$ and $$G_ af\in L^ p$$, where $G_ af(x)=(\int^{\infty}_{0}Ef(x,t)^ 2dt/t^{2a+1})^{1/2}.$ Also, if a is not an integer, the following variant of Ef, involving the Taylor polynomial Pf(y,x) of f at x can also be used instead of Ef $\Omega f(x,t)=\int_{| y| \leq t}| f(x+y)-Pf(y,x)| dy/t^ n.$ These results extend characterizations due to R. Strichartz in terms of vector valued means of differences, and of E. Stein in terms of Marcinkiewicz integrals.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26A16 Lipschitz (Hölder) classes 41A10 Approximation by polynomials
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##### References:
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