TITLE: Trig Integral Inequality
QUESTION [1 upvotes]: Show that if $f$ is Riemann integrable on $[a,b]$, then
$$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2\le(b-a)\int_{a}^{b}f^2(x)\ dx.$$
I know I need to use the Hölder Inequality in some form, but I am not sure how to rewrite the left side of the equation. I know I can rewrite the right side as $\left(\int_{a}^{b}\sin^2x+\cos^2x\ dx\right)\left(\int_{a}^{b}f^2(x)\ dx\right)$, but I'm struggling to invoke any established inequalities because of the addition. Thanks for any shared insights!

REPLY [0 votes]: $$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2=\Big|\int^b_a e^{ix} f(x)\,dx\Big|^2$$
An application of Cauchy-Bunyakovsky's inequality yields
$$\Big|\int^b_a e^{ix} f(x)\,dx\Big|\leq\Big(\int^b_a|e^{2ix}|^2\,dx\Big)^{1/2}\Big(\int^b_a|f(x)|^2\,dx\Big)^{1/2}=(b-a)^{1/2}\int^b_a|f(x)|^2\,dx$$


TITLE: Gaussian quadrature with weight function x^2
QUESTION [0 upvotes]: I would like to get the points and weights of Gaussian quadrature formulas for
$$
\int_{-1}^{+1} x^2 f(x)\;\text{d}x.
$$
Is this tabulated anywhere yet?

REPLY [0 votes]: For anyone interested, I've written orthopy for the derivation of Gaussian rules with new weight functions. The main example in the readme gives the points and weights
[-0.978228658146056992803938001123,
 -0.887062599768095299075157769304,
 -0.730152005574049324093416252031,
 -0.519096129206811815925725669458,
 -0.269543155952344972331531985401,
  0.269543155952344972331531985401,
  0.519096129206811815925725669458,
  0.730152005574049324093416252031,
  0.887062599768095299075157769304,
  0.978228658146056992803938001123]

[0.0532709947237135572432759986252,
 0.0988166881454075626728761840589,
 0.0993154007474139787312043384226,
 0.0628365763465911675266984722740,
 0.0190936733702070671592783399524,
 0.0190936733702070671592783399524,
 0.0628365763465911675266984722744,
 0.0993154007474139787312043384225,
 0.0988166881454075626728761840592,
 0.0532709947237135572432759986251]

for the 10-point Gaussian rule with the integral weight $x^2$.


TITLE: Explicit homeomorphism between an Infinite Cone and a one sheet hyperboloid
QUESTION [2 upvotes]: An infinite cone is given by the equation
$$B = \{(x,y,z) ∈ \mathbb R^3 : x^2 + y^2 −z^2 = 0\} $$
And a one sheeted hyperboliod is given by
$$C = \{(x,y,z) ∈ \mathbb R^3 : x^2 + y^2 −z^2 = 1\} $$
Are these two surfaces homeomorphic? And if yes, what are their explicit homeomorphisims?
I know both of these spaces are not compact, however I do not know how to check for other properties such as connectedness which are preserved under homeomorphisims.

REPLY [2 votes]: HINT: What happens if you remove the origin from the cone? Well, if they were homeomorphic, you'd remove the corresponding point from the hyperboloid. Now what?