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The lifting of polynomial traces revisited. (English) Zbl 1227.46026

Let \(\Theta\) be the rectangle \((-1,1)\times(0,1)\) and let \(H^1(\Theta)\) be the Sobolev space of order 1 on \(\Theta\). Let \(\Lambda\) be the lower edge \((-1,1)\times\{0\}\) of \(\Theta\) and let \(H^{1/2}(\Lambda)\) be the Sobolev space of order \(1/2\) on \(\Lambda\). The authors describe how to lift continuously elements of \(H^{1/2}(\Lambda)\) (“traces”) to elements of \(H^1(\Theta)\) in a polynomial preserving way: polynomial functions on \(\Lambda\) with degree \(N\) are lifted to polynomial functions on \(\Theta\) with degree at most \(N\) in each variable. The main tool is the Lebesgue differentiation theorem that yields a lifting from \(\Lambda\) into the equilateral triangle \(\mathcal T\) with base \(\Lambda\), combined with a homography that maps \(\mathcal T\cap\Theta\) onto \(\Theta\).
Then the authors describe how to lift continuously elements of \(H^{1/2}(\Lambda)\) that vanish in the endpoints of \(\Lambda\) (“flat” traces) to elements of \(H^1(\Theta)\) that vanish on the vertical edges of \(\Theta\) in a polynomial preserving way. Their method, a division by \((1-x)(1+x)\) followed by a lifting and by a multiplication with \((1-x)(1+x)\), requires the use of weighted Sobolev spaces; it has already been outlined in their note [C. R. Acad.Sci., Paris, Sér.I 315, No.3, 333–338 (1992; Zbl 0755.65103)]; see M. Ainsworth and L. Demkowicz [Math.Nachr.282, No.5, 640–658 (2009; Zbl 1175.46019)] for another method.
The article ends with two numerical illustrations. (1) The evaluation of the \(H^{1/2}(\Lambda)\)-norm of Legendre polynomials \(L_n\), which is known to grow like \(\sqrt{\log n}\). (2) The evaluation of the norm of the orthogonal projection \(\pi_N\) from \(L^2(\Lambda)\) onto the space of polynomial functions of degree at most \(N\) when acting on \(H^{1/2}(\Lambda)\): interpolation yields that it is bounded by \(CN^{1/4}\); numerical evidence leads the authors to conjecture that it grows in fact like \(\log N\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47B38 Linear operators on function spaces (general)
46B70 Interpolation between normed linear spaces
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