All Questions
23 questions
22
votes
2
answers
652
views
Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
- 1,209
21
votes
1
answer
3k
views
Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
- 4,034
11
votes
3
answers
2k
views
How can I simplify this sum any further?
Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...
- 213
10
votes
1
answer
594
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
- 13.8k
10
votes
2
answers
426
views
Density of the linear span of products of harmonic polymomials
Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...
- 577
3
votes
1
answer
137
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
- 463
3
votes
1
answer
102
views
Literature containing basic knowledge of homogeneous functions
Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
- 1,101
3
votes
0
answers
157
views
Growth of the constants from the Stone-Weierstrass Theorem
The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\...
- 10.6k
2
votes
1
answer
433
views
Stone-Weierstrass theorem: coefficients of approximating sequence bounded?
Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points ...
- 463
2
votes
2
answers
494
views
Polynomial approximation (Weierstrass theorem) with bounds
Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$.
Suppose $g = f$ at atmost finitely many points. Does there exist a ...
- 431
2
votes
2
answers
197
views
About preserving real-rootedness of multivariable polynomials
Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...
- 31
2
votes
1
answer
140
views
Polynomial approximation of continuous function with constraints
Consider the closed convex subset $\mathcal{F} = \{f \in C[0,1] : 0 \leq f \leq 1, f(0)=0, f(1)=1\}$. Consider the polynomial class $\mathcal{P} = \{p \text{ is a polynomial} : p(0)=0, p(1)=1, 0 \leq ...
- 431
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$
be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
2
votes
1
answer
74
views
Two variable polynomials that behave like Lagrange polynomials [closed]
Let us consider different points $z_i=(x_i,y_i)$ in the plane where $i=1,\cdots n$.
Q Do there exist two variable polynomials $P_i(x,y)$ with minimal degree such that $P_i(z_j)=\delta_{ij}$?
- 4,058
2
votes
1
answer
177
views
For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often
Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
- 1,209
2
votes
0
answers
122
views
Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc
Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
- 321
1
vote
2
answers
588
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
0
votes
1
answer
167
views
For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve
Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
- 1,209
0
votes
1
answer
152
views
When can two Cauchy transforms intersect?
Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...
- 1,893
0
votes
1
answer
281
views
Roots of linear combination of $x \sin x$
Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...
- 405
0
votes
0
answers
96
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
0
votes
0
answers
148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
-2
votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246