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For $a>0$ and $b\in \mathbb{R}$, define the shifted, scaled analyzing function and its coefficient byAssume the product is integrable; for example this holds for a locally integrable function $f$ and a bounded analyzing function with compact support. The bar is complex conjugation. This normalization preserves the L2 norm of the analyzing function. If $p$ is a polynomial of degree of a polynomial less than $q$, then $p(b+au)$ is a sum of monomials $u^r$ with $r<q$. Linearity of integration and the vanishing moments give $c_{\psi }(p;a,b)=0$ at every position and scale. Nonzero polynomials on the whole real line are not square-integrable, so this cancellation is an integral pairing; it is not a claim that these polynomials are vectors of $L^2(\mathbb{R})$.

Here is a quantitative reason that smooth regions yield small details. Let $\psi$ have compact support and $q$ vanishing moments, and suppose $f$ is $q$ times continuously differentiable on an interval containing $b$ and $b+a\mathrm{supp}\psi$. Subtract its degree-$(q-1)$ Taylor polynomial at $b$. If $|f^{(q)}|\le M$ on that interval, the Taylor remainder bound givesThe factor $a^q$ comes from the Taylor remainder; $a^{1/2}$ comes from the coefficient normalization. Thus the analyzing function measures the departure from the local polynomial trend, rather than the height of $f$ itself. The wavelet coefficient decay for Hölder functions uses the same argument with a Hölder-Taylor remainder bound. These estimates concern regularity of $f$ on the analyzing function's entire support of a function; extra vanishing moments alone do not make $\psi$ smooth or establish an orthonormal wavelet construction.^[ref]

If continuously differentiable $F_j,F$ converge uniformly on a compact interval and have uniformly bounded total variation of a function, then $\int b{F}_j^{\prime }\to \int b{F}^{\prime }$ for every continuous $b$. For continuously differentiable test functions this is integration by parts. Uniform approximation by polynomials extends the conclusion to continuous tests using the variation bound.

On $[0,1]$, the polynomials $p_n(x)=x^n/n$ converge in supremum norm to the zero function, since $\Vert p_n{\Vert }_{\infty }=1/n$. But $p_n^{\prime }(1)=1$ for every $n$, whereas the zero function has left derivative zero at that endpoint. Thus uniform convergence of function values does not imply uniform convergence of derivatives. This does not contradict uniform convergence of Bernstein polynomial derivatives, which has a particular approximation construction and additional differentiability hypotheses.

Every continuous real-valued function on a compact interval can be approximated by polynomials with arbitrarily small error everywhere on the interval. More precisely, if $a<b$ and $f:[a,b]\to \mathbb{R}$ is continuous, then for every $\epsilon >0$ there is a polynomial $p$ withThe error is measured in the supremum norm, so one approximation works at every point. The degenerate interval $a=b$ needs only a constant function. A change of variable $t=(x-a)/(b-a)$ reduces the nondegenerate case to $[0,1]$.^[ref]

This is useful because a polynomial is described by finitely many coefficients and can be evaluated with addition and multiplication. Its surprise is that the target need not be differentiable: the sharp corner of an absolute value function can be approximated by smooth polynomials. The Bernstein approximation of a corner below makes this visible. Approximating values, however, does not by itself approximate derivatives; Weierstrass approximation does not preserve derivatives gives a counterexample.

For a polynomial over a discretely valued field and a rational weight $t$, this coefficient-minimum function is multiplicative. Scale the variable by an element of valuation $t$ in a valued extension and reduce the initial terms; nonzero residue polynomials have nonzero product. For a monic polynomial factored into roots, $w_t(f)={\sum }_r\min (t,v(r))$. Its supporting lines determine the lower Newton polygon, giving the Newton polygon root valuation theorem and its multiplicities.

For a nonnegative weight $w$ and monic polynomials $p_j$ of degree $j$, this determinant is ${\prod }_i\sqrt{w(x_i)}{\prod }_{i<j}(x_j-x_i)$. Its square is the product of the weights times the squared Vandermonde determinant. For normalized weighted polynomials it differs by the product of their norm factors.

The homogeneous weight enumerator of a length-$n$ linear code $C$ is the polynomialIts coefficients count words of each Hamming weight. In this convention $W_C(1,1)=|C|$; some references interchange $s,t$.

A Zariski-closed set is a common zero set of polynomials. Arbitrary intersections and finite unions of such sets are again Zariski closed.

A zero of a cyclic code is a polynomial root of its generator polynomial, taken in a splitting field. Every codeword, represented as a polynomial, vanishes there. For a repeated-root cyclic code, the multiplicity of a root must also be retained: ordinary evaluations alone need not determine the code.

Collect the distinct nonzero values among the ${\lambda }_j$ as ${\mu }_1,\dots ,{\mu }_r$ with positive multiplicities $m_j$. The first $r$ power sums imply ${\sum }_j{m}_j{\mu }_j^k=0$ for $1\le k\le r$. The coefficient matrix is an invertible weighted Vandermonde matrix, so all $m_j$ vanish, a contradiction unless $r=0$. Its invertibility also follows because a polynomial ${\sum }_{k=1}^r{a}_k{z}^k$ vanishing at zero and all the ${\mu }_j$ has $r+1$ distinct roots and degree at most $r$.