The properties of general Fourier partial sums of functions with Lipschitz derivatives

In this paper, we investigate the Fourier partial sums with respect to a general orthonormal system (ONS) for functions belonging to the class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>L</m:mi> </m:msub> </m:math> {CL} , that is, functions whose derivatives belong to the Lipschitz class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Lip</m:mi> <m:mo>⁡</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {Lip1} . It is well known that the fact a function f (nonzero) belongs to a differential class does not guarantee the boundedness of its Fourier sums with respect to a general ONS. We establish certain conditions to be satisfied by the functions in the orthonormal system such that the Fourier partial sums of any function with derivatives from the Lipschitz class are bounded a.e. on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[0,1]} . The obtained results are optimal in a certain sense.