On Lebesgue summability for double series
| Autor: | M. J. Kohn |
|---|---|
| Rok vydání: | 1976 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 59:283-286 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1976-0415197-3 |
| Popis: | In [2] a two dimensional analogue of Lebesgue's theorem on differentiation of formally integrated trigonometric series was established. Here we show that a stronger analogue holds. 1. Let T: EFnEZ cn ein9 be a trigonometric series in one variable, with cn > 0. Let c() = 0 + *n n#0o ifl We say T is Lebesgue summable at 00 to sum s if X(8) has at 0o a first symmetric derivative with value s. That is, if '{X(0O + t) X(00 t)} = st + o(t) as t 0. The following result is well known (see [3, p. 322]). THEOREM A. Suppose cn = 0(1/n) as n -> oo and suppose T converges at 00 to finite sum s. Then T is Lebesgue summable at 00 to s. 2. We are concerned here with a two dimensional analogue of Theorem A for spherically convergent series. We denote points of T2 by x = (x1, x2) = tei9 and integral lattice points by n = (nl, n2). We write n * x = n1 x + n2x2and Inl = nn. Let L(x) be defined in a neighborhood of x0 E T2. We will say, see [2], that L has at xo a generalized first symmetric derivative with value s if L(x) is integrable over each circle I x xoI = t, for t small, and if (2.1) 1 L(xo + tei9)(cos 0 + sin 0)d0 = 'st + o(t) as t 0. If the limit in (2.1) exists only as t tends to 0 through a set having 0 as a point of density, we will say L(x) has at xo a generalized first symmetric approximate derivative. The following result was established in [2]. THEOREM B. Let T: E nEZ cn einfX be a double trigonometric series which converges spherically at xo to s, sl < so. Suppose the coefficients of T satisfy Received by the editors January 5, 1976. AMS (MOS) subject classifications (1970). Primary 42A92, 42A48, 42A24, 26A54. |
| Databáze: | OpenAIRE |
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