Function Spaces and Approximation of Continuous Functions

In this study, we have reviewed some function spaces and have approximated some continuous functions by polynomials for some application purpose. For example, considering tide at a point on the sea in the mean sea level is a continuous function with respect to time we have approximated it from some hourly observed data by Bangladesh Inland Water Transport Authority (BIWTA) through cubic spline interpolation . Also, the bottom topographic detail for the Bay of Bengal region covering the area between 150N and 230N Latitudes and 850E and 950E Longitudes is also approximated through inverse distance weighted interpolation from a figure with some doted depth contours. The obtained bathymetric data and generated tide can be used to solve shallow water equations for the prediction of water levels due to surge, tide or their interaction and many other purposes (such as safety of surface, sub-surface navigation, detecting something, submarine engineering etcetera) on this region. The method of compilation used in this study can be applied to compile the bathymetric data for any coastal region having a figure of doted depth contours.