Positive-definite function

In mathematics, a positive-definite function is, depending on the context, either of two types of function. Let be the set of real numbers and be the set of complex numbers. A function is called positive semi-definite if for any real numbers x1, ..., xn the n × n matrix is a positive semi-definite matrix. By definition, a positive semi-definite matrix, such as , is Hermitian; therefore f(−x) is the complex conjugate of f(x)). In particular, it is necessary (but not sufficient) that (these inequalities follow from the condition for n = 1, 2.) A function is negative semi-definite if the inequality is reversed. A function is definite if the weak inequality is replaced with a strong ( 0). If is a real inner product space, then , is positive definite for every : for all and all we have As nonnegative linear combinations of positive definite functions are again positive definite, the cosine function is positive definite as a nonnegative linear combination of the above functions: One can create a positive definite function easily from positive definite function for any vector space : choose a linear function and define . Then where where are distinct as is linear. Bochner's theorem Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0. The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure. In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Typically, n scalar measurements of some scalar value at points in are taken and points that are mutually close are required to have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an n × n matrix) is always positive-definite.