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Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0p. In this paper, the following result is proved:TheoremLet0N(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer $\psi \in L^1({\bf R}^d)$ , provided only that $\widehat{\psi}(0)=1$ and $\psi$ satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each $f \in H^1({\bf R}^d)$ , where $a_j$ is an arbitrary lacunary sequence (such as $a_j=2^j$ ) and the coefficients $c_{j,k}$ are local averages of f. This formula holds in particular if the synthesizer $\psi$ is in the Schwartz class, or if it has compact support and belongs to $L^p$ for some $1

In the general atomic setting of an unconditional basis in a (quasi-) Banach space, we show that representing the spaces of m-terms approximation as Lorentz spaces is equivalent to the verification of two inequalities (Jackson and Bernstein), and that the validity of these two properties is equivalent to the Temlyakov property. The proof is very direct and, especially, does not use interpolation theory. We apply this result to establish a representation theorem when the norm of the (quasi-) Banach space is given by a quadratic variation formula (thanks to a condition called the p-reverse inequality). This quadratic variation framework is in fact very rich and contains, as examples, the cases of Hardy spaces. We also consider the cases of "weighted" Hardy and Lebesgue spaces when the weight belongs to a Muckenhoupt class and the basis is a wavelet basis. This provides a new example of bases well adapted to approximation. [ABSTRACT FROM AUTHOR]