{"paper":{"title":"Range of certain convolution operators and reconstruction from local averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"P. Devaraj","submitted_at":"2018-05-30T05:19:31Z","abstract_excerpt":"For a compactly supported absolutely continuous measure $\\mu$ on ${\\mathbb{R}}^2$ having a density function equal to a finite linear combination of indicator functions of rectangles $\\left[a_{i}, b_{i}\\right]\\times \\left[c_{i}, d_{i}\\right],$ we analyse the range of the convolution operator $C_{\\mu}:C({\\mathbb{R}}^2)\\rightarrow C({\\mathbb{R}}^2)$ defined by $C_{\\mu}(f)=f\\star\\mu,$ where\n  $(f\\star \\mu)(x,y)=\\int_{{\\mathbb{R}}^2}f(x-s,y-t)d\\mu.$ It is shown that $C_{\\mu}$ maps the space of all continuous functions $C({\\mathbb{R}}^2)$ onto the space $C^{2*}({\\mathbb{R}}^2)=\\{f:{\\mathbb{R}}^2\\rig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.11810","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}