Abstract
We develop a wavelet-like representation of functions in \(L^{p}(\mathbb{R})\) based on their Fourier–Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the local smoothness of the target function. In the case of continuous functions, a similar expansion is given based on the values of the functions at arbitrary points on the real line. In the process, we give new proofs for the localization of certain kernels, as well as for some very classical estimates such as the Markov–Bernstein inequality.
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Acknowledgements
The research of HNM is supported in part by Grant W911NF-15-1-0385 from the US Army Research Office. We thank the editors for their kind invitation to submit this paper.
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Mhaskar, H.N. (2017). Local Approximation Using Hermite Functions. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_16
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DOI: https://doi.org/10.1007/978-3-319-49242-1_16
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