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On the rate of strong summability of double Fourier series. (English) Zbl 0936.42007
In the paper double trigonometric Fourier series of real-valued functions of two variables, \(2\pi \)-periodic in each variable and integrable in the Lebesgue sense with \(q\)-th power \((1\leq q<\infty)\) or continuous in the closed square \(Q=\langle -\pi ,\pi \rangle \times \langle -\pi ,\pi \rangle \) with the usual \(L^q\)-norm or sup-norm on \(Q\), are studied. Denote, for \(j,k=0,1,2,\ldots ,\gamma > -1\), \(\delta > -1\), \(\sigma ^{(\gamma ,\delta)}_{jk}\) the Cesàro \((C,\gamma ,\delta)\) means of the Fourier series of \(f\). Let us further denote \(L\) a summability method determined by an infinite functional sequence \(\{\alpha _k(r)/A(r)\}\), \(\alpha _k\geq 0\), \(A(r) = \sum ^{\infty }_{k=1} \alpha _k (r)\), regular in the following sense: if \(w_k\rightarrow w\) for \(k\rightarrow \infty \), then \(L\{w_k\} : = \frac 1{A(r)} \sum ^{\infty }_{k=1} \alpha _k (r) w_k\rightarrow w\) for \(r\rightarrow 1\), \(0<r<1\).
Seven theorems are proved, in which the strong means of Marcinkiewicz type with the Cesàro means of negative order in one of the variables instead of square partial sums are estimated by characteristics constructed on the basis of moduli of continuity. As an example we quote the first case of Theorem 1 for \(1<q<\infty \) and \(\gamma >0\):
Let \[ \sum ^{\infty }_{i=0} \Big (\sum ^{2(2^i-1)}_{k=2^i-1} (\alpha _k(r))^{\lambda } \Big (\omega _i \Big (\frac 1{k}\Big)\Big)^{\lambda p} \Big)^{\frac 1{\lambda }} \leq C_1 A(r)(1-r) \sum ^{\infty }_{k=0} \Big (\omega _i\Big (\frac 1{k}\Big)\Big)^p (i=1,2) \] for some \(\lambda >1\), \(\delta \) and \(p>0\) satisfying the condition \((1-\delta)p<1-\frac 1{\lambda }\). Then, if \((1-\delta) q<1-\frac 1{\lambda }\), we have, for \(0<r<1\), \[ \Big (L(\{\{\sigma ^{(\gamma ,\delta -1)}_{kk}-f\{^p_{L^q(Q)}\})\Big)^{1/p} \leq C_2 (1-r) \sum ^{\infty }_{k=0} r^k \Big (\omega ^p_1 \Big (\frac 1{k}\Big) + \omega _2^p \Big (\frac 1{k}\Big)\Big)^{\frac 1{p}}, \] where \(C_1,C_2\) are constants non depending on \(r\) and \(f\), and \(\omega _i(\rho)\) \((i=1,2)\) is the \(L_q(Q)\) integral modulus of continuity of \(f\) with respect to the \(i\)-th variable on the interval \(\langle 0,\rho \rangle \).
Reviewer: J.Fuka (Praha)
42B08 Summability in several variables
40F05 Absolute and strong summability (should also be assigned at least one other classification number in Section 40-XX)
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