We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an $L^2$ space weighted by the weight function of the continuous $q$-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a $q$-analog of the expansion of $e^{ixy}$ in Jacobi polynomials of argument $x$. We also outline a general procedure of finding integral representations for inverses of linear operators.
We discuss the simultaneous diagonalization of a family of commuting difference operators by Koornwinder's multivariable generalization of the Askey-Wilson polynomials. The operators constitute a complete set of quantum integrals for a difference version of the $n$-particle quantum Calogero-Sutherland system related to the root system $BC_n$.
We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients following from generalized Vandermonde determinant expansion. Our result can be viewed as a parametric version of a classical theorem of Liouville. We also give applications.
We establish good numerical estimates for a certain class of integrals involving sixfold products of Bessel functions. We use relatively elementary methods. The estimates will be used in the study of a sharp Fourier restriction inequality on the circle.
We investigate $g$-functions and Lusin's area type integrals related to certain multi-dimensional Dunkl and Laguerre settings. We prove that the considered square functions are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ into weak $L^1$.
In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis $ϕ_n(x)$ of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formulas for the considered integrals. Our approach, specified to elliptic case, is more effective than known recursive procedures for elliptic integrals. We also show that basic integrals enjoy symmetry coming from the action of the dihedral group $D_{M…
Generalized integral formulas involving the generalized modified k-Bessel function $J_{k,ν}^{c,γ,λ}\left( z\right) $ of first kind are expressed in terms generalized $k-$Wright functions. Some interesting special cases of the main results are also discussed
We aim to present two new generalized integral formulae involving product of generalized Struve function $\mathcal{W}_{p,b,c}\left( z\right)$, which are expressed in terms of the generalized Lauricella functions. The main results presented here, being the very general character, reduce to yield known and new integral formulae. Some special cases of our main results are also considered.
We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^ψ$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^ψ$ in the integral metric.
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed.
In this paper quantitative weighted matrix estimates for vector valued extensions of $L^{r'}$-Hörmander operators and rough singular integrals are studied. Strong type $(p,p)$ estimates, endpoint estimates, and some new results on Coifman-Fefferman estimates assuming $A_\infty$ and $C_p$ condition counterparts are provided. To prove the aforementioned estimates we rely upon some suitable convex body domination results that we settle as well in this paper.
We aim to introduce the generalized multiindex Bessel function $J_{\left( β_{j}\right) _{m},κ,b}^{\left( α_{j}\right)_{m},γ,c}\left[ z\right] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{\left( β_{j}\right) _{m},κ,b}^{\left( α_{j}\right)_{m},γ,c}\left[ z\right] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Ψ_{q}(z)$. The results presented here are of general in nature an…
We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{α,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{α_1} y_1^m+\cdots+x_n^{α_n} y_n^m)}f(y)dy, \end{equation} where the function $f$ is a Schwartz function. In this paper, the restriction theorem on $\mathbb{S}^{n-1}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures the restriction theorem hold.
In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \fracπ{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly rela…
The theory of self-reciprocal functions is applied to the study Mordell type integrals. We find two particular eigenfunctions of the double cosine Fourier transform and then use them to evaluate certain one- and two-dimensional Mordell type integrals in closed form. A reduction formula is given for a certain family of double Mordell integrals in terms of one-dimensional integrals.
In this paper we establish some (presumably new) interesting expressions for the composition of some well known fractional integral operators $ I^μ_{a+}, D^μ_{a+} $,$ I^{γ, μ}_{a+}$ and also derive an integral operator $\mathcal{H}^{w;m,n;α}_{a+;p,q;β}$ whose kernel involve the Fox's $H-$ function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition formulae which occur rather frequently in many problems of engineering and mathematical analysis but here we can mention only those…
On June 15, 1876, the Proceedings of the Royal Society of London published a paper by Mr. W. H. L. Russell entitled "On certain integrals". The paper starts with "The following are certain integrals which will, I hope, be found interesting". The rest of the paper is simply a list of 12 integrals. We provide a list of 12 integrals that we hope would be interesting.
In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize his results to the case that the phase is contained in a certain class of $C^{\infty}$ functions. The key in our analysis is a toric resolution of singularities in the above class of $C^{\infty}$ functions. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integ…
We obtain the $L^p$ decay of oscillatory integral operators $T_λ$ with certain homogeneous polynomial phase of degree $d$ in $(n+n)$-dimensions. In this paper we require that $d>2n$. If $d/(d-n)<p<d/n$, the decay is sharp and the decay rate is related to the Newton distance. In the case of $p=d/n$ or $d/(d-n)$, we also obtain the almost sharp decay, here "almost" means the decay contains a $\log(λ)$ term. For otherwise, the $L^p$ decay of $T_λ$ is also obtained but not sharp. A counterexample also arises in this paper to show that $d/(d-n)\leq p\leq d/n$ is not necessary to guarantee the sharp…
We give an exact result about the asymptotic limit of an oscillatory integral whose phase contains a certain flat term. Corresponding to the real analytic phase case, one can see an essential difference in the behavior of the above oscillatory integral.
Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ with $s\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $α\in(0,Q)$, the fractional integrals associated to the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$. In this article, we first introduce the Morrey space $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,κ}_{ρ,\infty}(\…
We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function $f$ along a collection of lines. Pick one such line, call it the $x$-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting $f$ to the line. Let $Ω_1$ be the interval where $f$ is supported, and $Ω_2$ be the interval where the Hilbert transform of $f$ can be computed using the Gelfand-Graev formula. The equation we study is $H_1 f=g|_{Ω_2}$, where $H_1…
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integ…
We investigate a family of integrals involving modified Bessel functions that arise in the context of neutrino scattering. Recursive formulas are derived for evaluating these integrals and their asymptotic expansions are computed. We prove in certain cases that the asymptotic expansion yields the exact result after a finite number of terms. In each of these cases we derive a formula that bounds the order at which the expansion terminates. The method of calculation developed in this paper is applicable to similar families of integrals that involve Bessel or modified Bessel functions.
One discusses a problem of asymptotical behavior for some operators in a general theory of pseudo differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with a help of the Fourier transform, Dirac mass-function and its derivatives, and well-known distribution related to the Cauchy type integral.
Given an increasing function $H:[0,1)\to [0,\infty)$ and $$ A_n(H):=\inf_{τ\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} (t_i-t)H^2(t)dt)^{1/2}, $$ where $\mathcal{T}_n:=\{τ=(t_i)_{i=0}^n: 0=t_0<t_1<...<t_n=1\}$, we characterize the property $A_n(H)\leq \frac{c}{\sqrt{n}}$, and give conditions for $A_n(H)\leq \frac{c}{\sqrt{n^β}}$ and $A_n(H)\geq \frac{1}{c\sqrt{n^β}}$ for $β\in (0,1)$, both in terms of integrability properties of $H$. These results are applied to the approximation of certain stochastic integrals.
The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was used successfully and systematically by Krattenthaler and Rao in their well known, very interesting research papers. The results are derived with the help of generalization of a quadratic transformation formula due to Kummer very recently obtained by Kim, et al. . Several identities including one obtained earlier by Krattenthaler and Rao follow special cases of…
Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_α(z) when α\ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^α(z) when α\ge -2, Jacobi polynomials P_n^{(α,β)}(z) when α\ge -1 and β\ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds., Marc…
