Basis and regularity properties of ( p , q )-trigonometric functions and the decay of ( p , q )-Fourier coefficients

The basis and regularity properties of the generalized trigonometric functionssin p , q andcos p , q are investigated. Upper bounds for the Fourier coefficients of these functions are given. Conditions are obtained under which the functionscos p , q generate a basis of every Lebesgue spaceL r (0,1) with1 < r < ∞ ; whenq is the conjugate ofp , it is sufficient to require thatp ∈[p 1 ,p 2 ], wherep 1 <2 andp 2 >2 are calculable numbers. A comparison is made of the speed of decay of the Fourier sine coefficients of a function in Lebesgue and Lorentz sequence spaces with that of the corresponding coefficients with respect to the functionssin p , q . These results sharpen previously known ones.