interpolaation

Interpolaation is a term often used as a misspelling of interpolation. Interpolation is a method of constructing new data points within the range of a discrete set of known data points. It involves selecting a function that matches the observed values at given abscissas, so that the function can estimate remaining values within the sample.

Common one-dimensional methods include linear interpolation, which connects adjacent points with straight lines; polynomial interpolation, which fits a single polynomial (using approaches such as Lagrange or Newton forms); and spline interpolation, which uses piecewise polynomials with smoothness constraints. Multidimensional interpolation extends these ideas to higher dimensions and scattered data, employing methods such as bilinear or bicubic interpolations on grids, radial basis functions, or kriging in geostatistics.

The mathematical goal is to find a function f such that f(xi) = yi for a set of known pairs (xi, yi). If the xi are distinct and there are n points, there exists a polynomial of degree at most n−1 that interpolates the data, though high-degree polynomials can exhibit oscillations outside the sample range (a phenomenon known as the Runge phenomenon). For noisy data or larger data sets, local or smoothing approaches are often preferred to avoid overfitting.

Applications of interpolation span numerical analysis, data visualization, signal and image processing, computer graphics, and geographic information systems, where estimating intermediate values is essential for reconstruction, analysis, or display.