An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series
Abstract
:1. Introduction
2. Approximation Using the Dirichlet Operator
3. The Sansone Estimates
3.1. Estimate of
3.2. Estimate of
3.3. Estimate for
3.4. Estimate of
3.5. Estimate of
- (i)
- The term
- (ii)
- Arguing as in (i), we have:
- (iii)
- The mean square on of:
- (iv)
- The method of (ii) applied to the estimation of the square mean on , of:
- (v)
- The square mean, on , of:
4. The Proof of Theorem 1
5. An Example
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
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Huang, M.L.; Kerman, R.; Spektor, S. An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series. Mathematics 2018, 6, 64. https://doi.org/10.3390/math6040064
Huang ML, Kerman R, Spektor S. An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series. Mathematics. 2018; 6(4):64. https://doi.org/10.3390/math6040064
Chicago/Turabian StyleHuang, Mei Ling, Ron Kerman, and Susanna Spektor. 2018. "An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series" Mathematics 6, no. 4: 64. https://doi.org/10.3390/math6040064
APA StyleHuang, M. L., Kerman, R., & Spektor, S. (2018). An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L2(ℝ) by a Partial Sum of Its Hermite Series. Mathematics, 6(4), 64. https://doi.org/10.3390/math6040064