On the weak behaviour of partial sums of Legendre series. (English) Zbl 0534.42005
The author proves two theorems on convergence of Legendre expansions \(\sum^{\infty}_{n=0}a_ nP_ n(x)\sim f(x);\) let \(S_ N(f)(x)\) denote the N-th partial sum of the series. The first result states that for a set E with characteristic function \(\chi_ E(x)\) and with Lebesgue measure m(E) it yields \(m\{x:\quad | S_ N(\chi_ E)(x)|>y\}\leq(C/y^ 4)(m(E)),\)-1\(\leq x\leq 1\); \(y>0\) (C independent of N, and E). On the other side, in general for \(f\in L^ 4(-1,1)\), there exists no constant C, independent of N, and f, so that \(m\{x:\quad | S_ N(f)(x)|>y\}\leq(C/y^ 4)\cdot \| f\|^ 4_ 4,\)- 1\(\leq x\leq 1\). Earlier H. Pollard [Trans. Am. Math. Soc. 62, 387- 403 (1947; Zbl 0040.322)] showed that in case \(4/3<p<4 \| S_ N(f)\|_ p\leq C_ p\cdot \| f\|_ p\) is true. This last inequality is false when \(p\leq 4/3\) or \(p\geq 4 (p<4/3\) or \(p>4:\) H. Pollard (loc. cit.), \(p=4/3\) or \(p=4:\) J. Newman and W. Rudin [Proc. Am. Math. Soc. 3, 219-222 (1952; Zbl 0046.294)]).
Reviewer: H.Schwinn
MSC:
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
References:
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