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Prove the series converges almost everywhere

Question: Given Lebesgue integrable $f: mathbb{R}rightarrow [0,infty)$, prove the following series converges almost everywhere on $mathbb{R}$:$$varphi(x) = lim_{krightarrow infty} sum_{t=-k}^k f(t+x)$$ Attempt: Towards a contradiction suppose...

Mathematics Asked by Christopher Rose on 2 years ago

Prove that the functional in $C_c^0(Omega)$ is a Radon measure

Let $Omega subset mathbb{R}^n$ be an arbitrary open set and $(x_n)_{n inmathbb{N}} subset Omega$ a sequence. Let $(a_n)_{n inmathbb{N}} subset mathbb{C}$ be a sequence such that...

Mathematics Asked on 2 years ago

How can I evaluate ${lim_{hto 0}frac{cos(pi + h) + 1}{h}}$?

I'm supposed to evaluate the following limit using the cosine of a sum and one of the "special limits" which are ${lim_{xto 0}frac{sin(x)}{x}=1}$ and ${lim_{xto 0}frac{1-cos(x)}{x}=0}$. The limit...

Mathematics Asked by DCdaKING on 2 years ago

Estimate $f(b)$ using Taylor Expansion for $f'(x) = cos(x^2)$

I am using Taylor Expansion for the following problem, but for some reason I am getting wrong solutions from a program I am running it on. Can someone please help...

Mathematics Asked by brucemcmc on 2 years ago

If $f ∈ C^∞(M)$ has vanishing first-order Taylor polynomial at $p$, is it a finite sum of $gh$ for $g, h ∈ C^∞(M)$ that vanish at $p$?

This is 11-4(a) in Lee's "Introduction to Smooth Manifolds": Let $M$ be a smooth manifold with or without boundary and $p$ be a point of $M$. Let...

Mathematics Asked by Fred Akalin on 2 years ago

$ sum_{n=1}^infty csc^2(omegapi n)= frac{A}{pi} +B $

$$ sum_{n=1}^infty csc^2(omegapi n)= frac{A}{pi} +B $$ if $omega =-frac{1}{2}+frac{sqrt{3}}{2}i$ find $frac{A^2}{B^2}$My Attempt$$ sum_{n=1}^infty csc^2(omegapi n)= sum_{n=1}^infty csch^2(iomegapi n)= 4sum_{n=1}^infty big(e^{pi n big( frac{i}{2} +...

Mathematics Asked by hwood87 on 2 years ago

Show that $f$ is a strong contraction when $f$ is continuously differentiable.

Let $f: [a,b] to R$ be a differentiable function of one variable such that $|f'(x)| le 1$ for all $xin [a,b]$. Prove that $f$ is a...

Mathematics Asked on 2 years ago

Decomposition of a linear operator to a partially orthogonal operator and a semi-definite self-adjoint operator

$DeclareMathOperator{A}{mathscr{A}}$$DeclareMathOperator{B}{mathscr{B}}$$DeclareMathOperator{C}{mathscr{C}}$$DeclareMathOperator{kernel}{mathrm{Ker}}$$DeclareMathOperator{diag}{mathrm{diag}}$$DeclareMathOperator{span}{mathrm{span}}$$DeclareMathOperator{real}{mathbb{R}^2}$$DeclareMathOperator{rank}{text{rank}}$ The question is:Let $A$ be a linear operator on the $n$-dimensional Euclidean...

Mathematics Asked by Zhanxiong on 2 years ago

Fourier expansions of Eisenstein series as a Poincare series for the Fuchsian group

In Miyake's book, Modular Forms, Ch 2.6, thm 2.6.9, there is a statement which relate to Fourier expansion of the Eisenstein series. Let $Gamma$ be a Fuchsian group, ...

Mathematics Asked by LWW on 2 years ago

Is eigenvalue multiplied by constant also an eigenvalue?

Let $A$ be an $n × n$ matrix. If $lambda$ is an eigenvalue of $A$ and $c$ is a nonzero scalar, then $clambda$ is...

Mathematics Asked by Ruby Cho on 2 years ago

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