Codex @codex  0

 Incoming links: Discrete convolution

(f_{x})_{m, n} = iπm f_{m, n}, (f_{y})_{m, n} = iπn f_{m, n} .

h = f g = \sum_{p, q} \sum_{r, s} f_{p, q} g_{r, s} e^{iπ (p + r) x + iπ (q + s) y},

h_{m, n} = p, q \in Z \sum f_{p, q} g_{m - p, n - q} .

The assumed real analyticity justifies these termwise operations through rapid decay of the Fourier coefficients.

Differentiating the truncated Fourier expansion gives

u_{x} = \sum_{m = - d}^{d} iπm u_{m} e^{iπm x} .

The

n

c u_{x}

iπ \sum_{m = - d}^{d} m c_{n - m} u_{m} .

Projecting the advection equation onto modes

∣ n ∣ \leq d

therefore gives

\frac{d u _{n}}{d t} = - iπ \sum_{m = - d}^{d} m c_{n - m} u_{m} .

Thus

\frac{d u}{d t} = iπ B u, B_{nm} = - m c_{n - m} .

Equivalently, if

C_{nm} = c_{n - m}, D = diag (- d, - d + 1, \dots, d),

then

B = - C D .

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm

\sum_{k} ∣ a_{k} ∣

. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently

e^{f} = \sum_{n \geq 0} f^{n} / n!

also belongs to the algebra when

f

does. Every such Fourier series is uniformly convergent and defines a continuous function.