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        Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

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        Best constant for Hölder inequality in Lorentz spaces

        It's well known (and proved by R. O'neil) that there is a version of H?lder's inequality for Lorentz spaces, namely $$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...
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        When can a function be made positive by averaging?

        Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...
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        2answers
        108 views

        What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

        $\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$
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        1answer
        128 views

        How do you prove the validity of this formula for $H(n)$? [closed]

        I'm looking for a proof of the identity $$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$ There is a generalization of this formula for $...
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        Reference request: Oldest calculus, real analysis books with exercises?

        Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there. Edit. Unsolved exercises ...
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        40 views

        Convex functions and infinite convex combinations

        Let $f: D\rightarrow \mathbb R$ be a convex function on a convex subset in $\mathbb R^n$. Let $t_i>0$ with $\sum_{i=1}^\infty t_i=1$ and $x_\in D$ be such that the series $\sum_{i=1}^\infty t_i x_i ...
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        1answer
        78 views

        Are bilinear sparse bounds for local operators trivial?

        I'm thinking about a recent result by M. Lacey which says that a dyadic spherical maximal function satisfies a sparse bilinear bound. To be precise, define the unit scale dyadic spherical maximal ...
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        0answers
        18 views

        Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2

        Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation: Do both $w_1$ and $w_2$'s ...
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        2answers
        165 views

        Integral equality of 1st intrinsic volume of spheroid

        Computations suggest that $$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$ The question ...
        5
        votes
        1answer
        197 views

        Finding an asymptotic solution for a first order ODE

        Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...
        3
        votes
        2answers
        276 views

        how to calculate the following integral related to Chebyshev polynomials

        Chebyshev polynomials of the second kind $V_n(x)$ can be defined as $$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$ or through the recurrence relation $$V_{n+1}=xV_n-V_{n-1}, V_0=1, V_1=...
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        3answers
        163 views

        A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

        Is there an analytical solution satisfying these 3 equations with non-constant z? $$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...
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        1answer
        186 views

        The discrete Hardy-Littlewood-Sobolev inequality

        Let $p>1$, $q>1$, $0<\lambda<1$ be such that $\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that $(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$. It is known ([1,2,3]...
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        0answers
        122 views

        Integral of $C^\infty$ Analog of Unit Step Function

        The function $$ f(x)\ :=\ e^{1-\frac{1}{\sqrt{1-x^2}}} $$ has the properties that $\frac{d^nf}{dx^n}(\pm 1)=0$, $\frac{d^n}{dx^n}\left(\sqrt{1-x^2}-f(x)\right) = 0$ at $x=0$, and its integral $F(...

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        山西福彩快乐十分钟
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