Function spaceIn mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Vector space#Function spaces Let be a vector space over a field and let be any set.
InterpolationIn the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.
Multivariate interpolationIn numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).
Polynomial interpolationIn numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.
Hölder's inequalityIn mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if 1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in Lp(μ) and g is in Lq(μ), then the pointwise product fg is in L1(μ).
