Theta Functions from an Analytic Viewpoint

Abstract

Suppose that we have a measure space (X, dx). We recall that this is a pair consisting of a non-empty set X and a (countably additive) measure dx on X. We shall denote a general point of X by x and consider C-valued functions Φ(x), or simply Φ, on X. For 1 ≦ p < ∞ we shall denote by L^p(X) the set of L^p-functions on X, i.e., the set of measurable functions Φ on X such that |Φ(x)| p is integrable with respect to dx. The set L^p(X) forms a vector space over C. Moreover, if we define the L^p-norm ∥Φ∥_p of Φ as $\|\Phi\|_p=({\mathop\int\limits_X}\mid\Phi(x)\mid^p\ dx)^{1/p},$ this has the usual properties of a norm, and it is invariant under complex conjugation. In particular, if we take ∥Φ₁ − Φ₂∥_p the “distance” of two “points” Φ₁, Φ₂ of L^p(X), we get a metric space. We tacitly identify Φ₁, Φ₂ if ∥Φ₁ − Φ₂∥_p = 0. The metric space L^p(X) is complete. If Φ is a bounded (continuous) function on X, we shall denote by ∥:∥_∞ the uniform norm of Φ, i.e., the supremum of ∣Φ(x)∣ for all x in X. We shall consider mostly the space L²(X) and its subspaces. We recall that, for every Φ₁, Φ₂ in L²(X), Φ₁(x) Φ₂(x) is integrable with respect to dx, hence the scalar product: $(\Phi_1,\ \Phi_2)={\mathop\int\limits_X}\Phi_1(x)\overline{\Phi_2(x)}dx$ is well defined. This is C-linear in Φ₁ and its complex conjugate is (Φ₂, Φ₁). In other words, L²(X) forms a Hubert space. We shall denote the L ²-norm simply by ∥ ∥; we have the Schwarz inequality: $\mid(\Phi_1,\ \Phi_2)\mid{\mathop <\limits_=}\|\Phi_1\|\|\Phi_2\|.$ if (Y, dy) is another measure space, we can define the product measure dx ⊗ dy, or simply dx dy, on the product X × Y. If Φ(x, y) is an L¹-function on X × Y with respect to dx dy, we have the Fubini theorem: $\mathop\int \limits_{X \times Y} \Phi(x, y)dx dy = {\mathop\int \limits_X}({\mathop\int \limits_Y} \Phi(x, y)dy)dx = {\mathop\int \limits_Y}({\mathop\int \limits_X} \Phi(x, y)dx)dy.$