Represent fx y k constant 0 lessthanorequalto x lessthanor

Represent f(x, y) = k = constant (0 lessthanorequalto x lessthanorequalto a,0 lessthanorequalto y lessthanorequalto b) by a double Fourier series.

Solution

For a periodic function on the domain [0,a]x[0,b] , the double Fourier series is the expansion of the function f(x) in terms of A(x) B(y) , where A and B are trigonometric functions in x and y with periods a and b.

If f(x,y) = k is a constant, then the Fourier series expansion consists only of the constant (coefficient ) term, in view of the orthogonal property of the constant function wrt sin and cos functions

So the required double Fourier series consists of only one term k