By G. H. Hardy
There might be few textbooks of arithmetic as recognized as Hardy's natural arithmetic. considering the fact that its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have became initially in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the elemental principles of the differential and indispensable calculus, of the homes of limitless sequence and of alternative subject matters related to the suggestion of restrict.
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P ( . t) 2q-2 n ! n q, §1l The Associated Spaces y~(q) 55 §11 The Associated Spaces y~ (q) There are two aspects of the subject of this section. We may see it as a process to generate orthonormalized bases in a rational way, but we may also regard it in view of symmetry properties, related to the isotropical group J(q, eq). We follow the historical way and generalize Laplace's first integral. 1) a(q) = eq . 2) homogeneous harmonics, depending on 71(q-l). 3) is a homogenous and harmonic polynomial of degree n, and consequently an element of y~(q).
Theorem 1: Each space Yn(q), q 2: 3, n 2: 0 is the orthogonal direct sum of the associated spaces Y;;'(q) Proof: First, we show that the associated spaces in Yn(q) are mutually orthogonal. For 0 :0:::; k,m :0:::; n, Yk(q - l;'),Ym (q -1;·) and the scalar products < , >(q) ; < , >(q-1) we have (§11. ) = < Yk, Ym >(q-1) 1 >(q) A~(q; t) A;;:(q;t)(l- t2)~dt Thus Y! 1 Y;;' for k -=I- m. Moreover A~ is a bijection of Yk (q - 1) onto y~(q). 9) u=O N(q,n)u n l+u (1 - U)q-1 l+u 1 (1 - U)q-2 1 - u §11 The Associated Spaces y~ (q) 57 Since we are dealing with two linear spaces of equal dimension N(q, n), the assertion is proved.
Let us assume we have an orthonormal basis Y n ,l, Y n ,2, ... , Yn,N of Yn(q). We can find a system 0:1, 0:2, ... 2) Y n ,2(o:d ;k = 1, ... ,N-l 32 1. The General Theory can be constructed recursively. 4) Yn(~) = L CjYn,j(~) j=l and the coefficients are uniquely determined if we presecibe the values in at, ... 3) without knowing the elements Yn,j. This leads to Definition 1: A system of N(q, n) points al,"" aN on Sq-1 is called regular of degree n if the N x N determinant is positive. The system is called singular if the determinant is zero.