On the biharmonic and allied operators (Q762397)

Consider the class L of all \(2\pi\)-periodic complex-valued Lebesgue integrable functions in the interval \(<-\pi,\pi >\). For \(f\in L\) write \[ \omega (f;x,\delta) = \sup_{0<| h|\leq\delta} \left\{\frac{1}{| h|} \left| \int^{h}_{0} (f(x+t)-f(x)) dt \right| \right\}. \] A point x at which \(\int^{h}_{0} (f(x+t) -f(x))dt = o(h)\) as \(h\to 0\) is called a weak Lebesgue point of f. If \(a_ n\) and \(b_ n\) are the Fourier coefficients of f, the biharmonic and allied operators, denoted by \(B_ r\), \(\tilde B_ r\) \((0<r<1)\) are defined by the formulae \[ B_ r[f](x) = a_0/2 + \sum^{\infty}_{k=1} \{1+(k/2)(1-r^ 2)\} r^ k(a_ k\cos kx+b_ k\sin kx) \tag{1} \] \[ \tilde B_ r[f](x) = \sum^{\infty}_{k=1} \{1+(k/2)(1-r^ 2)\} r^ k(a_ k\sin kx-b_ kc os kx). \tag{2} \] The author studies the order of pointwise convergence of the means (1) and (2) as \(r\to 1\)- at Lebesgue points for functions of the class L and also obtains estimates for the derivatives \((d/dx)B_ r[f](x)\) and \((d/dx)\tilde B_ r[f](x)\). A typical result is as follows: For every weak Lebesgue point x of f and all \(r\in <1/2,1),\) \[ | B_ r[f](x)-f(x)| \leq \frac23(1-r)^2\omega(f;x,\pi)+ \] \[ (8+\pi^5)(1-r)^3 \int^{\pi}_{1-r} \frac{\omega (f;x,t)}{t^ 4}dt + \frac{\pi^ 3}{2} (1-r)^2 \int^{\pi}_{1-r}\frac{\omega (f;x,t)}{t^2} dt. \] It has been shown that this estimate cannot be essentially improved.