zbMATH — the first resource for mathematics

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

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.)
Full Text: EuDML