{"paper":{"title":"An estimate of the root mean square error incurred when approximating an $f \\in L^2({\\mathbb{R}})$ by a partial sum of its Hermite series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Mei Ling Huang, Ron Kerman, Susanna Spektor","submitted_at":"2017-08-29T00:57:48Z","abstract_excerpt":"Let $f$ be a band-limited function in $L^2({\\mathbb{R}})$. Fix $T >0$ and suppose $f^{\\prime}$ exists and is integrable on $[-T, T]$. This paper gives a concrete estimate of the error incurred when approximating $f$ in the root mean square by a partial sum of its Hermite series.\n  Specifically, we show, for $K=2n, \\quad n \\in Z_+,$\n  $$\n  \\left[\\frac{1}{2T}\\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\\right]^{1/2}\\leq \\left(1+\\frac 1K\\right)\\left(\\left[ \\frac{1}{2T}\\int_{|t|> T}f(t)^2dt\\right]^{1/2} +\\left[\\frac{1}{2T} \\int_{|\\omega|>N}|\\hat f(\\omega)|^2d\\omega\\right]^{1/2} \\right) +\\frac{1}{K}\\left[\\frac{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03039","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"}