Relating smoothness to expressions involving Fourier coefficients or to a Fourier transform

Coefficients of expansion of a function by trigonometric, algebraic and spherical harmonic orthogonal polynomials are related to the smoothness of that function. Hausdorff–Young type and Hardy–Littlewood type inequalities will be utilized. Expressions involving the Fourier transform of a function are also related to the measure of smoothness of that function.