The role of the relative voltage and phase for frequency coupling in a dual-frequency capacitively coupled plasma

Frequency coupling in multifrequency discharges is a complex nonlinear interaction of the different frequency components. An alpha-mode low pressure rf capacitively coupled plasma operated simultaneously with two frequencies is investigated and the coupling of the two frequencies is observed to greatly influence the excitation and ionization within the discharge. Through this, plasma production and sustainment are dictated by the corresponding electron dynamics and can be manipulated through the dual-frequency sheath. These mechanisms are influenced by the relative voltage and also the relative phase of the two frequencies.

Whitney's type theorems for infinite dimensional spaces

It is proved that for any $f$ is an element of $C^k(L,R)$, where k is a natural number and L is a closed linear subspace of a nuclear Frechet space $X$, the function $f$ can be extended to a function of class $C^{k-1}$ defined on the entire space $X$. It is also proved that for any $f$ is an element of $C^k(L, R)$, where $k$ is a natural number of infinity and L is a closed linear subspace of a dual $X$ of a nuclear Frechet space, the function $f$ can be extended to a function of class $C^k$ defined on the entire space $X$. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.