Abstract
The classical Bernstein estimate for absolutely convergent Fourier series and its multivariate extension are improved. Analogous results for absolutely convergent expansion by Legendre polynomials and by spherical harmonic polynomials are given. For expansion by a Fourier series it follows that the present result implies the classical result. Moreover, there are many functions that are shown to have an absolutely convergent Fourier series using the present estimate while not satisfying the conditions of the previous one.
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