Asymptotic behavior of trigonometric series with \(O\)-regularly varying quasimonotone coefficients
Let \(\{a_n\}_{n=1}^\infty\) be a null sequence of complex numbers, and \(f(x):=\sum_{n=1}^\infty a_n\cos nx,\) \(g(x):=\sum_{n=1}^\infty a_n\sin nx,\) and \(h(x):=\sum_{n=1}^\infty a_n e^{inx}.\) Let \[ \begin{alignedat}{2} a_n&\sim An^{-\alpha}l(n) &&\qquad\text{as}\quad n\to\infty, \tag{1}\\ f(x)&\sim Ax^{\alpha-1}l(1/x)\Gamma(1-\alpha)\sin\alpha\pi/2 &&\qquad\text{as} \quad x\to 0+,\tag{2}\\ g(x)&\sim Ax^{\alpha-1}l(1/x)\Gamma(1-\alpha)\cos\alpha\pi/2 &&\qquad\text{as} \quad x\to 0+, \tag{3}\\ h(x)&\sim Ax^{\alpha-1}l(1/x)\Gamma(1-\alpha)e^{\pi i(1-\alpha)/2} &&\qquad\text{as}\quad x\to 0+.\tag{4}\end{alignedat} \] Very general conditions on \(\{a_n\}\) and \(l(x)\) are considered under which (1) yields (2)--(4) and, conversely, those under which (2)--(4) yield (1). By this, various known results due to Hardy, Rogosinski, Zygmund, and others are considerably extended.
scientific article; zbMATH DE number 1539395