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

Abstract

The basis and regularity properties of the generalized trigonometric functions sin p , q and cos p , q are investigated. Upper bounds for the Fourier coefficients of these functions are given. Conditions are obtained under which the functions cos p , q generate a basis of every Lebesgue space L r (0,1) with 1 < r < ∞ ; when q is the conjugate of p , it is sufficient to require that p ∈[ p 1 , p 2 ], where p 1 <2 and p 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 functions sin p , q . These results sharpen previously known ones.