Abstract
We study the multipliers of multiple Fourier series
for a regular system on anisotropic Lorentz spaces.
In particular, the sufficient conditions for a sequence of
complex numbers {λk}k∈Zn in order to make it a multiplier
of multiple trigonometric Fourier series from Lp[0; 1]n to
Lq[0; 1]n , p > q. These conditions include conditions Lizorkin
theorem on multipliers.
In this work, we are concerned that transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate.
We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .