Functions on the Plane as Combinations of Powers of Distances to Points

Some real functions on the plane can be expressed as a linear combination of powers of distances to certain points. 1. Introduction In this paper we construct a function basis to express different powers of the distance between a point and the origin. Namely, for two given natural numbers and we find points in the plane such that the function can be expressed as linear combination of powers of distances to those points (see formula (3.15) below), both the linear combination and the given function having the same domain in the plane (for the notion of special function that we use, see Section 3.1). One of the important observations in the theory of real functions is that series expansions lead to certain interesting numbers and functions (e.g., Fourier coefficients, -functions). Namely, usually series representations can be constructed explicitly in some model spaces of sections of various function bundles over appropriate analytic manifolds. (By considering functions on the plane as maps : , the graphs of these functions are the sections described by the map : ; ( ) ( , ( )), which defines 2-sub-manifolds on .) (for analytic functions on the plane ; see Section 2.2). We denote by the subspace of real analytic functions. It is well known that any polynomial of degree can be written exactly as a linear combination of powers of the right number of terms of the form . We realize a series representation in the space of analytic functions on the plane . Consider a point in the plane with coordinates ( ). The distance used here is the Euclidean distance. Denote We prove the following theorem. Theorem 1.1. Consider two natural numbers and . There are numbers and points , such that Remark 1.2. The numbers and the points can be determined exactly. Corollary 1.3. The linear combinations of , , , can be expressed similarly. Remark 1.4. An analogous theorem holds for the sine function. Denote by the maximum integer less than or equal to . Consider the function 2. Proof of Theorem 1.1 We will denote by the Legendre polynomial of degree . The following formula is formula (8.921) of [1]: Consider the set of . Consider a subset (not necessarily proper) of such set. We will work solely with analytic functions on . Consider the series . Divide this number by . By formula (2.1), the following formula holds: We use the formula of the product of two series and : The right side of formula (2.2) becomes Let us change the order of addition with respect to the indices and : The following formula is formula (8.911) of [1]: where Substituting (2.6) in (2.5), Let us change the