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They may serve as optimized basis functions in interpolation or approximation, or as shape functions in meshless methods for PDE solving. Their norm is useful for proving upper bounds for convergence rates of interpolation in Sobolev spaces\n \n \n $$H_2^m(\\mathbb {R}^d)$$<\/jats:tex-math>\n \n \n \n H<\/mml:mi>\n 2<\/mml:mn>\n m<\/mml:mi>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and this paper gives the correct rate\n \n \n $$m-d\/2$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n -<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n that arises as convergence like\n \n \n $$h^{m-d\/2}$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n \n m<\/mml:mi>\n -<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for interpolation at meshwidth\n \n \n $$h\\rightarrow 0$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or a blow-up like\n \n \n $$r^{-(m-d\/2)}$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \n -<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n -<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for norms of compactly supported functions with support radius\n \n \n $$r\\rightarrow 0$$<\/jats:tex-math>\n \n \n r<\/mml:mi>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In Hilbert spaces with infinitely smooth reproducing kernels, like Gaussians or inverse multiquadrics, there are no compactly supported functions at all, but in spaces with limited smoothness, compactly supported functions exist and can be optimized in the above way. The construction is described in Hilbert space via projections, and analytically via trace operators. Numerical examples are provided.\n <\/jats:p>","DOI":"10.1007\/s10444-026-10293-9","type":"journal-article","created":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T07:14:40Z","timestamp":1772522080000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal compactly supported functions in Sobolev spaces"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7828-4355","authenticated-orcid":false,"given":"Robert","family":"Schaback","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,3]]},"reference":[{"key":"10293_CR1","volume-title":"Sobolev spaces","author":"R Adams","year":"1975","unstructured":"Adams, R.: Sobolev spaces. Academic Press, New York, N.Y. (1975)"},{"key":"10293_CR2","volume-title":"Sobolev spaces","author":"R Adams","year":"2003","unstructured":"Adams, R., Fournier, J.J.F.: Sobolev spaces. 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