Let$A$and$B$be positive numbers and$m$and$n$positive integers,$m<n$. Then there is for complex valued functions φ on$R$with sufficient differentiability and boundedness properties a representationwhere$v$_{$1$}and$v$_{$2$}are bounded Borel measures with$v$_{$1$}absolutely continuous, such that there exists a function φ with ∣φ^{(n)}∣ ⩽$A$and ∣φ∣ ⩽$A$on$R$and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.

The product ϕ_{λ}^{(α,β)}(t_{1})ϕ_{λ}^{(α,β)}(t_{2}) of two Jacobi functions is expressed as an integral in terms of ϕ_{λ}^{(α,β)}(t_{3}) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.

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In this paper, we describe all (2,3)-torus structures of a highly symmetric 39-cuspidal degree 12 curve. A direct computer-aided determination of these torus structures seems to be out of reach. We use various quotients by automorphisms to find torus structures. We use a height pairing argument to show that there are no further structures.