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Register a new userThe Fourier Transforms of the Chebyshev and Legendre Polynomials
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Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called unified method in the interior of a convex polygon provides an example of the applicability of these analytic expressions.
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We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and expand on an alternative approach based on a theorem of Schoenberg.
Given $n$ polynomials $p_1, dots, p_n$ of degree at most $n$ with $|p_i|_infty le 1$ for $i in [n]$, we show there exist signs $x_1, dots, x_n in {-1,1}$ so that [Big|sum_{i=1}^n x_i p_iBig|_infty < 30sqrt{n}, ] where $|p|_infty := sup_{|x| le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(sqrt{n})$ for the Chebyshev polynomials $T_1, dots, T_n$, and can be seen as a polynomial analogue of Spencers six standard deviations theorem.
A word $sigma=sigma_1...sigma_n$ over the alphabet $[k]={1,2,...,k}$ is said to be {em smooth} if there are no two adjacent letters with difference greater than 1. A word $sigma$ is said to be {em smooth cyclic} if it is a smooth word and in addition satisfies $|sigma_n-sigma_1|le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation.
We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based method for the Wiener-Hopf equation due to Henery, expressing it in terms of the Hilbert transform, and computing the latter in a more sophisticated way with sinc functions. We then generalise our method to the Fredholm equation reformulating it as two coupled Wiener-Hopf equations and solving them iteratively. We provide numerical tests and open-source code.
A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugation transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or several (possibly infinitely many) ones. Our analysis yields a few by-products, such as results related to positive definite operators, and to functional equations and inclusions involving monotone operators.
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