An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L 2 (ℝ) by a Partial Sum of Its Hermite Series

Mei Ling Huang, Ron Kerman and Susanna Spektor
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Mei Ling Huang: Department of Mathematics and Statistics, Brock University, St. Catharines, L2S 3A1, ON, Canada
Ron Kerman: Department of Mathematics and Statistics, Brock University, St. Catharines, L2S 3A1, ON, Canada
Susanna Spektor: Department of Mathematics and Statistics, Brock University, St. Catharines, L2S 3A1, ON, Canada

Mathematics, 2018, vol. 6, issue 4, 1-18

Abstract: Let f be a band-limited function in L 2 ( R ) . Fix T > 0 , and suppose f ′ 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. Specifically, we show, that for K = 2 n , n ∈ Z + , 1 2 T ∫ − T T [ f ( t ) − ( S K f ) ( t ) ] 2 d t 1 / 2 ≤ 1 + 1 K 1 2 T ∫ | t | > T f ( t ) 2 d t 1 / 2 + 1 2 T ∫ | ω | > N | f ^ ( ω ) | 2 d ω 1 / 2 + 1 K 1 2 T ∫ | t | ≤ T f N ( t ) 2 d t 1 / 2 + 1 π 1 + 1 2 K S a ( K , T ) , in which S K f is the K -th partial sum of the Hermite series of f , f ^ is the Fourier transform of f , N = 2 K + 1 + 2 K + 3 2 and f N = ( f ^ χ ( − N , N ) ) ∨ ( t ) = 1 π ∫ − ∞ ∞ sin ( N ( t − s ) ) t − s f ( s ) d s . An explicit upper bound is obtained for S a ( K , T ) .

Keywords: Hermite functions; Fourier–Hermite expansions; Sansone estimates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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