On the summability of random Fourier–Jacobi series

Nonnegative Fourier–Jacobi Coefficients and Some Classes of Functions

In this note we improve a theorem of I. P. Natanson by proving that if $f'(x)$ belongs to a Lipschitz class of order greater than or equal to $\frac{1}{2}$ on $[ - 1,1]$ then the Fourier–Jacobi series of $f(x)$ converges uniformly to $f(x)$ on $[ - 1,1]$.

Uniform Convergence of Fourier–Jacobi Series

Strong and Weak Weighted Convergence of Jacobi Series

In the case -1/2 -1/2, conditions are obtained in terms of the matrix of a linear method of summability and, among others, an antipole condition, which ensure the convergence of the linear means of the Fourier–Jacobi means of an integrable function at the Lebesgue point x=1. In the case -1/2<α<1/2 and -1β ≤ 1/2, it is proved that the linear means of the Fourier–Jacobi series converge at the Lebesgue point x=1, without any additional antipole condition.

A special class of Jacobi series and some applications

Weak behaviour of Fourier-Jacobi series

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

The inverse quantum scattering problem for the perturbed Bessel equation is considered. A direct and practical method for solving the problem is proposed. It allows one to reduce the inverse problem to a system of linear algebraic equations, and the potential is recovered from the first component of the solution vector of the system. The approach is based on a special form Fourier–Jacobi series representation for the transmutation operator kernel and the Gelfand–Levitan equation which serves for obtaining the system of linear algebraic equations. The convergence and stability of the method are proved as well as the existence and uniqueness of the solution of the truncated system. Numerical realization of the method is discussed. Results of numerical tests are provided revealing a remarkable accuracy and stability of the method.

S.A. Agahanov and G.I. Natanson [1] established lower and upper bounds for the Lebesgue functions L(α,β) n (x) of Fourier–Jacobi series on the interval [−1, 1]. The bounds differ from each other only in a constant factor depending on Jacobi parameters α and β, so their result is of final character. The aim of this paper is to extend their estimation for the weighted Lebesgue functions L(α,β),(γ,δ) n (x) using Jacobi weights with parameters γ and δ. We shall also give sufficient conditions with respect to α, β, γ and δ for which the order of the weighted Lebesgue functions is log (n + 1) on the whole interval [−1, 1].

Sampling theorems are one of the basic tools in information theory. The signal function f whose band–region is contained in a certain interval can be reconstructed from their values f ðxkÞ at the sampling points fxkg. We obtain analogues of this theorem for the cases of the Fourier–Jacobi series, the complex sphere S 1 c and the complex semisimple Lie groups. And as an application of these formulae, we show a version of the sampling theorem for the Radon transform on the complex hyperbolic space.

Generalized Lebesgue constants for the Fourier–Jacobi sums and the convergence of Fourier–Jacobi series in the L 1,A,B spaces are investigated.

We use the method of Bruinier–Raum to show that symmetric formal Fourier–Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.

Let \(P_k^{(\alpha ,\beta )} \) be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators \({\mathcal{U}}_n^\Lambda :C\left[ { - 1,1} \right] \to C\left[ { - 1,1} \right]\)generated by a multiplier matrix \(\Lambda = \left\{ {\lambda _k^{\left( n \right)} } \right\}\) defined by the following relations: $$f \sim \sum\limits_{k = 0}^\infty {a_k P_k^{\left( {\alpha ,\beta } \right)} } ,{\mathcal{U}}_n^\Lambda f \sim \sum\limits_{k = 0}^\infty {\lambda _k^{\left( n \right)} } a_k P_k^{\left( {\alpha ,\beta } \right)} ,$$ and $${\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right) = \mathop {sup}\limits_{y \in \left[ { - 1,1} \right]} \mathop {sup} \limits_{\left\| f \right\| \leqslant 1} \left| {\mathcal{U}_n^\Lambda f\left( y \right)} \right|.$$ In the case |α| = |β| = 1/2, we prove the following statements for the Jacobi polynomials (these statements are similar to known results for the trigonometrical system). Consider the cases $$(1)\;\;\;\alpha = \beta = - {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {k/n} \right);$$ $$\left({2} \right)\;\;\;\alpha = \beta = {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {\left( {k + 1} \right)/n} \right);$$ and $$\left( {3} \right)\;\;\;\alpha = - \beta = \pm {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {\left( {k + {1}/{2}} \right)/n} \right).$$ Under some conditions on a function ϕ, the values \(\mathop {\sup }\limits_{n \in {\mathbb{N}}} {\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right)\) and \(\mathop {\lim }\limits_{n \to \infty } {\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right)\) equal $$\frac{2}{\pi }\int_{0}^\infty {\kern 1pt} {\kern 1pt} \left| {\int_{0}^\infty {\varphi \left( t \right)\cos zt\;dt} } \right|dz\;\;\;\left( {case\left({1} \right)} \right)$$ and $$\frac{2}{\pi }\int_0^\infty z \left| {\int_0^\infty {t\varphi \left( t \right)\sin zt\;dt} } \right|dz\;\;\left( {cases\;\left({2} \right)\;and\;\left( \right)} \right).$$ In addition, we show that for the Fourier–Legendre summation methods (α = β = 0) generated by the multiplier function \(\lambda _k^{\left( n \right)} = \varphi \left( {k/n} \right)\), the limit and supremum of the sequence of Lebesgue constants may differ. Bibliography: 11 titles.

The article focuses on the problem of basis property for the Jacobi polynomial system \(P_n^{\alpha ,\beta }(x)\) in the weighted variable exponent Lebesgue space \(L^{p(x)}_\mu ([-1,1])\). The sufficient, and in a certain sense, necessary conditions on the variable exponent p = p(x) > 1 ensuring the uniform boundedness of Fourier-Jacobi sums \(S_n^{\alpha ,\beta }(f)\) (n = 0, 1, …) with − 1 < α, β < −1∕2 in \(L^{p(x)}_\mu ([-1,1])\) are obtained.KeywordsJacobi polynomialsFourier-Jacobi sumsWeighted Lebesgue space with variable exponentMathematics Subject Classification (2010)42C1046E30