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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.

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