The intrinsic mode functions (IMFs) arise as basic modes from the application of the empirical mode decomposition (EMD) to functions or signals. In this procedure, instantaneous frequencies are subsequently extracted from the IMFs by the simple application of the Hilbert transform, thereby providing a multiscale analysis of the signal's nonlinear phases. The beauty of this redundant representation method is in its simplicity and extraordinary effectiveness in many important and diverse settings. A fundamental issue in the field is to better understand these demonstrated qualities of the EMD procedures and the elementary modes they produce. For example, it is easily observed that when an EMD procedure is applied to the sum of two arbitrary IMFs, the original modes are rarely reproduced in the generated collection of IMFs. An interesting question from a representation point of view may be stated as follows: for any given sufficiently smooth function and fixed n ≥2, when is it possible to represent the function as a sum of (at most) n intrinsic modes? A more interesting question is whether such a decomposition is possible when the extracted modes are constructed from a common formulation of the intrinsic properties of the function being analysed. We provide an answer to these questions for a relaxed version of IMFs, called weak IMFs , which has been shown to be characterized in terms of eigenfunctions of Sturm–Liouville operators. The objective of this study is to further extend that analogy to the relationship between sums of weak IMFs and coupled Sturm–Liouville systems . The construction of this decomposition also provides a guide to an alternate characterization of the instantaneous frequency and bandwidth.