Igor Novikov

Haar series and linear operators

1. Preliminaries. 2. Definition and Main Properties of the Haar System. 3. Convergence of Haar Series. 4. Basis Properties of the Haar System. 5. The Unconditionality of the Haar System. 6. The Paley Function. 7. Fourier-Haar Coefficients. 8. The Haar System and Martingales. 9. Reproducibility of the Haar System. 10. Generalized Haar Systems and Monotone Bases. 11. Haar System Rearrangements. 12. Fourier-Haar Multipliers. 13. Pointwise Estimates of Multipliers. 14. Estimates of Multipliers in L1. 15. Subsequence of the Haar System. 16. Criterion of Equivalence of the Haar and Franklin Systems in R.I. Spaces. 17. Olevskii System. References. Index.

Fourier-Haar Multipliers

In this section we investigate multipliers with respect to the H.s. A sequence λ n , k , (n, k) ∈ Ω generates the operator

Generalized Haar Systems and Monotone Bases

\Lambda (\sum\limits_{(n,k) \in \Omega } {{a_{n,k}}x_n^k} ) = \sum {{\lambda _{n,k}}{c_{n,k}}x_n^k}

Convergence of Haar Series

(1) on the polynomials with respect to the H.s. Such operators are said to be multipliers. Recall that the norm of Λ from L p into L p (L p ) is denoted by ‖Λ‖p,q (‖Λ‖ p ). The main result of Chapter 5 (Corollary 5.8) may be formulated in the following way. If , then (2)

The Haar system and martingales

Theorem 3.2 shows that the H.s. forms a basis in L p , 1 ≤ p < ∞. This statement may be generalized.

Reproducibility of the Haar system

The purpose of this chapter is to describe monotone bases in r.i. spaces. If any contractive projection P satisfying the condition Pk (0,1) = k (o,1) is a conditional expectation, then such description can be given in terms of generalized Haar systems. We start in section 10.a with the characterization of r.i. spaces with the above mentioned property.

Haar System Rearrangements

The Franklin system is an orthonormal system obtained by the Gram-Schmidt orthonormalization from the Faber-Schauder system Recall that

Pointwise Estimates of Multipliers

One of the main propeties of the H.s. is that it forms a basis in C, L p (1 ≤ p 1) is discontinuous. Therefore if x ∈ C[0,1], then the convergence S n x to x is meant in L ∞.