𝐿^{𝑝} Bernstein estimates and approximation by spherical basis functions

Abstract

The purpose of this paper is to establish <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sphere. In particular, the Bernstein inequality estimates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm of the function itself. An important step in its proof involves measuring the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stability of functions in the approximating space in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript p"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript upper P"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>P</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm. Finally, we give a new characterization of Besov spaces on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sphere in terms of spaces of SBFs.