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Part of 1963 Miklós Schweitzer

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Miklos Schweitzer 1963_8

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9/19/2008
Let the Fourier series a02+∑k≥1(akcos⁡kx+bksin⁡kx) \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx) of a function f(x) f(x) be absolutely convergent, and let ak2+bk2≥ak+12+bk+12  (k=1,2,...) . a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ . Show that 1h∫02π(f(x+h)−f(x−h))2dx  (h>0) \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0) is uniformly bounded in h h. [K. Tandori]
trigonometryfunctionintegrationsearchreal analysisreal analysis unsolved