Gegenbauer Series Research Articles

Comment on the charge-renormalization effects of quartic scalar self-interactions

The charge-renormalization effects of quartic scalar interactions are calculated in a general gauge theory at the three-loop level using the Gegenbauer series expansion technique. The result agrees with a previous calculation by Curtright.

A method for Feynman diagram evaluation

The evaluation of multiloop dimensionally-regularized Feynman diagrams by expansion of (momentum space) propagators in Gegenbauer series is described.

On the singularities of Gegenbauer (ultraspherical) expansions

The results of Gilbert on the location of the singular points of an analytic function f ( z ) f(z) given by Gegenbauer (ultraspherical) series expansion f ( z ) = Σ n = 0 ∞ a n C n μ ( z ) f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^\mu (z) are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter’s results on distributions given by Legendre series: f ( z ) = Σ n = 0 ∞ a n C n 1 / 2 ( z ) f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z) . The singularities of the analytic representation of f ( z ) f(z) are compared to those of the associated power series g ( z ) = Σ n = 0 ∞ a n z n g(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}{z^n} . The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.

Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series

Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series

Weighted norm inequalities for the Hardy maximal function

The principal problem considered is the determination of all nonnegative functions, $U(x)$, for which there is a constant, C, such that \[ \int _J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int _J {|f(x){|^p}U(x)dx,} } \] where $1 < p < \infty$, J is a fixed interval, C is independent of f, and ${f^ \ast }$ is the Hardy maximal function, \[ {f^ \ast }(x) = \sup \limits _{y \ne x;y \in J} \frac {1}{{y - x}}\int _x^y {|f(t)|dt.} \] The main result is that $U(x)$ is such a function if and only if \[ \left [ {\int _I {U(x)dx} } \right ]{\left [ {\int _I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqq K|I{|^p}\] where I is any subinterval of J, $|I|$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when $p = 1$ or $p = \infty$, a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.

Applications of a Class of Singular Partial Differential Equations to Gegenbauer Series which Converge to Zero

Applications of a Class of Singular Partial Differential Equations to Gegenbauer Series which Converge to Zero