Relating coefficients of expansion of a function to its norm

Relations of the coefficients of the expansion of a function by orthogonal sequence with the norm of that function are given. The results contain Hausdorff–Young and Hardy–Littlewood type inequalities. The many applications that show the use of the results and their values are crucial. Many of the applications are for expansion by orthogonal polynomials on domain D and weight w  . For w=1w=1 the results are given for polytopes and other bounded convex domains. For the ball, simplex and the cube the weights dealt with are Jacobi-type. For D=RD=R, R+R+, and Rd(d>1) we give results using Freud (including Hermite) weights, Laguerre weights and Hermite weights respectively. Expansions by trigonometric polynomials on the torus and by spherical harmonic polynomials on the sphere are also dealt with.