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On Fourier coefficients with respect to the Haar system of functions of $$L_ p$$ spaces. (Russian) Zbl 0574.42005
In this remarkable paper the Haar-Fourier coefficients of $$L_ p$$ functions are investigated. It is worth while quoting some of the results. If $$c(f)=\{c_ m(f)\}$$ are the Haar-Fourier coefficients of f, set $\| f\|^*\!_ p=\lim_{n\to \infty}\{2^{- n}\sum^{2^{n+1}-1}_{m=2^ n}L_ m^{p/2}(c(f))\}^{1/p}$ $$L_ m(c(f))=c_ 0(f)$$ if $$m=0$$, $$=\sum^{n+1}_{i=0}a^ 2_{[m2^{-i}]}2^{(n-i)^+}$$ if $$m=1,2,...$$. One of the main results of the paper is that for $$1<p<\infty$$ the norms $$\| f\|_ p$$ and $$\| f\|^*\!_ p$$ are equivalent. This yields a necessary and sufficient condition for a sequence to be the Haar-Fourier sequence of an $$L_ p$$ function without referring to $$L_ p$$ norms (as is done in Marcinkiewicz celebrated inequality). The author also gives another reformulation of this result in the form $\| f\|_ p\sim \{| c_ 0(f)|^ p+\sum^{\infty}_{m=1}(L_ m^{p/2}(c(f))- L^{p/2}_{[m/2]}(c(f)))\}^{1/p}.$ Then he turns to simpler necessary or sufficient conditions e.g. as $c_ 1\sum | c_ m(f)|^ p(V_ m(m+1))^{p/2-1}\leq \| f\|^ p_ p\leq \sum | c_ m(f)|^ p(m+1)^{p/2}$ (1$$<p<2$$, $$V_ m\nearrow$$, $$\sum (V_ m(m+1))^{-1}<\infty)$$. These are shown to be best possible. In the third paragraph the order of approximation of functions by the partial sums of their Haar-Fourier series is studied, finally the fourth paragraph is devoted to the properties of special Haar series. E.g. using a result of P. L. Ul’yanov it is proved that if (*) $$\max_{2^ n\leq m<2^{n+1}}| a_ m| \leq C\min_{2^{n-1}\leq m<2^ n}| a_ m|,$$ and a sequence of partial sums $$\{$$ $$\sum^{q_ n}_{1}a_ m\chi_ m(t)\}$$ is bounded on a set of positive measure, then $$\{a_ m\}$$ is the Haar-Fourier coefficient sequence of a function belonging to every $$L_ p$$, $$p>1$$. Furthermore, if $$N_ 1$$ is any subsequence of the natural numbers and if the left- hand side of (*) is replaced by $$\max_{2^ n\leq m<2^{n+1},m\not\in N_ 1}| a_ m|$$, then the assertion fails.
Reviewer: V.Totik

##### MSC:
 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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