Double Fourier Series

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Suppose $f(x,y)$ and $g(x,y)$ are two real valued functions. They could for example be Eigenfunctions of an Eigenvalue problem, like they are encountered in solving PDEs. Since they are dependent on two variables $x$ and $y$ they shall be defined on the rectangle $R:$ $0 \leq x \leq L_1$ and $0 \leq y \leq L_2$. We define the inner product \begin{align} (f\mid g)=\int_0^{L_1} \int_0^{L_2} f(x,y)g(x,y) \, dx \, dy \end{align} Two functions $f_{mn}$ and $g_{ij}$ are orthogonal if

\begin{align} (f_{mn} \mid g_{ij})=\int_0^{L_1} \int_0^{L_2} f_{mn}(x,y)g_{ij}(x,y) \, dx \, dy=\left\{\begin{array}{ll} 0 & \text{for } m \not= n \hspace{0.2cm} \text{or} \hspace{0.2cm}i \not= j \\ 1 & \text{for } m=n \hspace{0.2cm} \& \hspace{0.2cm} i=j \\ \end{array}\right. \tag{1} \end{align} provided the functions are already normalised.

Now consider we want to expand an arbitrary function $\phi(x,y)=\phi(\mathbf{x})$ into a

Double Fourier Series \begin{align} \phi(\mathbf{x})= \sum_n A_n v_n(\mathbf{x}) \end{align}

Let us now choose a specific example e.g. $v(\mathbf{x})=v(x,y)=sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_1} \right)$ then we have \begin{align} \phi(\mathbf{x})= \sum_n A_n v_n(\mathbf{x})= \sum_{lm} A_{lm} sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_2} \right) \end{align} How can we determine the coefficients $A_{lm}$? First we recognize, that for the chosen $v(x,y)$ the orthogonality relation for $(f_{lm}\mid f_{ij})$ like stated in (1) is satisfied. This means that we can multiply the Double Fouerier Series by $v(\mathbf{x})$, integrate over $0 \leq x \leq L_1$, $0 \leq y \leq L_2$ and make use of orthogonality \begin{align} \int \phi(\mathbf{x})v_k(\mathbf{x}) d\mathbf{x}= \sum_n A_n \int v_n(\mathbf{x})v_k(\mathbf{x}) d\mathbf{x}= \sum_{lm} A_{lm} \int_0^{L_1} \int_0^{L_2} sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_2}\right) sin \left(i\frac{\pi x}{L_1} \right) sin \left(j\frac{\pi y}{L_2}\right) \,dx \, dy = \left\{\begin{array}{ll} 0 & \text{for } m \not= n \hspace{0.2cm} \text{or} \hspace{0.2cm}i \not= j \\ \frac{L_1L_2}{2^2} & \text{for } m=n \hspace{0.2cm} \& \hspace{0.2cm} i=j \\ \end{array}\right. \end{align} We find the coefficients \begin{align} A_{lm}=\frac{\int_0^{L_1} \int_0^{L_2} \phi(x,y)sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_2}\right)dx\, dy }{\int_0^{L_1} \int_0^{L_2} \left| sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_2}\right) \right|^2 dx\, dy}=\frac{4}{L_1L_2}\int_0^{L_1} \int_0^{L_2} \phi(x,y)sin \left(l\frac{\pi x}{L_1} \right) sin \left(m\frac{\pi y}{L_2}\right) \, dx \, dy \end{align}

Another approach is to treat the first sum as separate constant $B_m$ \begin{align} \phi(\mathbf{x})= \sum_{m=0} \underbrace{\left( \sum_{l=0} A_{lm} sin \left(l\frac{\pi x}{L_1} \right) \right)}_{B_l} sin \left(m\frac{\pi y}{L_2} \right) \end{align} This gives an ordinary Fourier Series \begin{align} \phi(\mathbf{x})= \sum_{m=0} B_l \, sin \left(m\frac{\pi y}{L_2} \right) \end{align} for which we can determine $B_m$ \begin{align} B_l= \int_0^{L_1} \phi(x,y) \, sin \left(m'\frac{\pi y}{L_2} \right) dx \end{align} and through that $A_{ml}$ \begin{align} B_l=\sum_{l=0} A_{lm} \, sin \left(l\frac{\pi x}{L_1} \right) \end{align} with \begin{align} A_{lm} =\int_0^{L_2} B_l \, sin \left(l'\frac{\pi x}{L_1} \right) \end{align}

For the a General Fourier Series in the Region $D$ \begin{align} \phi(\mathbf{x})= \sum_n A_n v_n(\mathbf{x}) \end{align} where $v_n(\mathbf{x})$ are orthogonal Eigenfunctions, the general coefficients are \begin{align} A_n=\frac{(\phi \mid v_n)}{(v_n \mid v_n)}=\frac{\iiint_D \phi (\mathbf{x})v_n(\mathbf{x}) \, d\mathbf{x}}{\iiint_D \left| v_n(\mathbf{x}) \right|^2 \, d\mathbf{x}} \end{align}

Further Reading:

  • Harris F. Daivs - Fourier Series and Orthogonal Functions
  • Schaums Outlines - Fouerier Analysis
  • Walter Strauss - Partial Differential Equations
  • G. B. Folland - Fouerier Analysis and its Applications