  This paper proposes a novel, rigorous and simple Fourier-transformation
approach to study resonances in a perfectly conducting slab with finite number
of subwavelength slits of width $h\ll 1$. Since regions outside the slits are
variable separated, by Fourier transforming the governing equation, we could
express field in the outer regions in terms of field derivatives on the
aperture. Next, in each slit where variable separation is still available, wave
field could be expressed as a Fourier series in terms of a countable basis
functions with unknown Fourier coefficients. Finally, by matching field on the
aperture, we establish a linear system of infinite number of equations
governing the countable Fourier coefficients. By carefully asymptotic analysis
of each entry of the coefficient matrix, we rigorously show that, by removing
only a finite number of rows and columns, the resulting principle sub-matrix is
diagonally dominant so that the infinite dimensional linear system can be
reduced to a finite dimensional linear system. Resonance frequencies are
exactly those frequencies making the linear system rank-deficient. This in turn
provides a simple, asymptotic formula describing resonance frequencies with
accuracy ${\cal O}(h^3\log h)$. We emphasize that such a formula is more
accurate than all existing results and is the first accurate result especially
for slits of number more than two to our best knowledge. Moreover, this
asymptotic formula rigorously confirms a fact that the imaginary part of
resonance frequencies is always ${\cal O}(h)$ no matter how we place the slits
as long as they are spaced by distances independent of width $h$.
