bibkey: itoh2015weightedapproximation authors: Kentaro Itoh, Ryozi Sakai, and Noriaki Suzuki year: 2015 title: The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight doi: 10.1155/2015/706930 url: https://arxiv.org/abs/1311.3337v2 claim: The published weighted Favard inequality, combined with the classical Brascamp–Lieb inequality and the existing complete-prime form comparison, supplies a conditional polynomial approximation of the whole original fixed-gap spectral subspace. It supplies no matrix sign or RH conclusion. strata_touched: [] license: bibliographic-reference-only triage: anchor

Weighted polynomial approximation in the original theta form

Reused source statements

Itoh–Sakai–Suzuki, arXiv:1311.3337v2, Proposition 6.2, gives, for $W=e^{-Q}\in \mathcal{F}(C^2+)$, $1\le p\le \infty$, and absolutely continuous $g$ with $g^{\prime }W\in L^p$,

$$E_{p,n}(W,g):=\underset{\mathrm{deg}r\le n}{\inf }\Vert (g-r)W{\Vert }_p\le C_J\frac{a_n}{n}\Vert g^{\prime }W{\Vert }_p.\text{\hspace{1em}(WA1)}$$

Here $a_n$ is the Mhaskar–Rakhmanov–Saff number defined in its introduction. The source attributes this Favard inequality to Sakai–Suzuki, On the Favard-type inequalities for exponential weights, Theorem 1, and their earlier Theorem 6.1. It is reused, not reproved. The published constant $C_J=C(W,p)$ is not numerically evaluated here. For complex $g$, apply the real estimate to its real and imaginary parts; in $L^2$ this keeps the same constant.

Lubinsky, arXiv:math/0701099v1, Definition 3.11 and Theorem 3.12, pp. 17–19, place this problem in classical Erdős-weight approximation. Definition 7.5 and Theorem 7.6, pp. 46–47, give the relevant weight hypotheses and warn of the extra $\sqrt{T(a_n)}$ factor in a polynomial derivative bound. A value-only Jackson bound is not substituted for a simultaneous derivative bound.

The classical Brascamp–Lieb variance inequality is stated as (1.3) in Carlen–Cordero-Erausquin–Lieb, arXiv:1106.0709v2, and attributed there to Brascamp–Lieb, DOI 10.1016/0022-1236(76)90004-5. For the probability density proportional to $e^{-2Q}$ with $Q^{\prime \prime }\ge \kappa >0$, it gives

$$\Vert (f-{\mu }_W(f))W{\Vert }_2^2\le \frac{1}{2\kappa }\Vert f^{\prime }W{\Vert }_2^2,{\mu }_W(f)=\frac{\int f{W}^2}{\int W^2}.\text{\hspace{1em}(WA2)}$$

Real and imaginary parts give the same assertion for complex $f$. These are ordinary approximation and log-concavity results, not RH-strength inequalities for the original jump form.

Same-form comparison and an admissible auxiliary weight

Use the unchanged normalized measure $d\nu =\rho dx$, $\rho =2\Phi \cosh (x/2)$, and minimal even form norm $\Vert h{\Vert }_{\mathcal{F}}^2=\Vert h{\Vert }_{\nu }^2+D(h)$ from the complete weighted Fourier construction. Its (WF2) already retains every prime power. Put

$${\gamma }_j=C_j/18,L_{\rho }={\gamma }_1/2+1/4,H_{\rho }=({\gamma }_2+{\gamma }_1^2)/2+1/8,K=c_0+2c_1{L}_{\rho }^2.$$

The original coefficient bounds (WC1) give, for $a=(\log \sqrt{\rho }{)}^{\prime }$,

$$|a(x)|\le L_{\rho }{e}^{4|x|},|a^{\prime }(x)|\le H_{\rho }{e}^{8|x|}.$$

Indeed the second logarithmic derivative uses the ${\gamma }_2{e}^{6|x|}$ and ${\gamma }_1^2{e}^{8|x|}$ terms, and the $\log (2\cosh (x/2))$ second derivative is at most $1/4$. For an even smooth error $e$ with the displayed finite weighted norms, (WF2), applied to $v=\sqrt{\rho }e$, gives

$$\Vert e{\Vert }_{\mathcal{F}}^2\le K\Vert e^{4|x|}\sqrt{\rho }e{\Vert }_2^2+2c_1\Vert \sqrt{\rho }{e}^{\prime }{\Vert }_2^2.\text{\hspace{1em}(WA3)}$$

Finiteness puts $v$ in $H^1$, so (WF2) also puts $e$ in the actual minimal domain. No identification with a maximal energy domain is used.

To check the approximation theorem’s hypotheses on an explicit weight, write $r=|x|$, $b=13/2-\pi >0$, and set

$$Q(x)=\frac{\pi }{2}(e^{2r}-1-2r)-b\log \cosh r,W(x)=e^{-Q(x)},\kappa =3\pi -13/2>0.\text{\hspace{1em}(WA4)}$$

This $Q$ is even and $C^2$, with $Q(0)=Q^{\prime }(0)=0$. For $r>0$,

$$Q^{\prime \prime }(r)=2\pi e^{2r}-b{\mathrm{sech}}^{2}r\ge \kappa ,Q^{\prime \prime \prime }(r)=4\pi e^{2r}+2b{\mathrm{sech}}^{2}r\tanh r>0.$$

Thus $Q$ is nonnegative and strictly convex. Increasing $Q^{\prime \prime }$ gives $r{Q}^{\prime }(r)\ge 2Q(r)$. The function $T(r)=r{Q}^{\prime }(r)/Q(r)$ has limit 2 at zero and is asymptotic to $2r$ at infinity; these limits and continuity make it quasi-increasing. The positive continuous ratio $Q{Q}^{\prime \prime }/(Q^{\prime }{)}^2$ tends to $1/2$ at zero and 1 at infinity, and is bounded above and below away from zero. These facts verify each condition of $\mathcal{F}(C^2+)$ in §2 of the Itoh–Sakai–Suzuki source. No $C^3$ or $C^4$ weight-class theorem is applied.

Every original theta-series summand is positive on $r\ge 0$. Retaining its first summand, and using (WC1) for the upper bound, gives

$$(4{\pi }^2-6\pi )e^{5r-\pi e^{2r}}\le \rho (r)\le 2C_0{e}^{5r-\pi e^{2r}}.$$

Since $r-\log 2\le \log \cosh r\le r$, put

$$S=\sqrt{2C_0}{2}^b{e}^{-\pi /2},A_W=\frac{e^{\pi /2}}{\sqrt{4{\pi }^2-6\pi }}.$$

Then, on the whole real line,

$$e^{4r}\sqrt{\rho }\le SW,W/\sqrt{\rho }\le A_W{e}^{4r}.\text{\hspace{1em}(WA5)}$$

The auxiliary weight changes only the approximation estimate. The original measure, covariance, jump form, prime powers and spectral projection remain unchanged.

One polynomial controls value and derivative errors

For a compact smooth even complex $f$, let $r_n$ be its derivative’s $L^2(W^{2}dx)$ orthogonal projection onto the polynomials of degree at most $n$. The even weight and odd $f^{\prime }$ make $r_n$ odd. Put

$$p_n^0(x)={\int }_0^x{r}_n(t)dt,p_n=p_n^0+{\mu }_W(f-p_n^0).$$

The map $f\mapsto p_n$ is complex linear, $p_n$ is even with degree at most $n+1$, and ${\mu }_W(f-p_n)=0$. Apply (WA1) to $g=f^{\prime }$ and (WA2) to this same error. Both norms are finite, since $f$ is compact smooth and $p_n$ has polynomial growth. With

$$C_{\mathcal{F}}=S\sqrt{K/(2\kappa )+2c_1},$$

(WA3)–(WA5) give

$$\Vert f-p_n{\Vert }_{\mathcal{F}}\le C_{\mathcal{F}}{C}_J\frac{a_n}{n}\Vert f^{\prime \prime }W{\Vert }_2.\text{\hspace{1em}(WA6)}$$

All polynomials here belong to the same minimal even domain by the existing polynomial cutoff argument. The projection and the primitive are a parameter adaptation of the published approximation and variance bounds. They are not a new generic density, derivative or prime-distribution theorem.

Apply to the whole existing low-spectral family

Keep the original $P_{\epsilon }$ and Fourier cutoff $p_N$ of (WF8)/(FF2). Choose the spatial cutoff from one fixed translated smooth profile, with the same support and first-derivative properties as (FF1)–(FF2), and $\Vert {\chi }_R^{\prime \prime }{\Vert }_{\infty }\le C_{\chi }$ independent of $R$. Such a profile can be obtained by smoothing a unit-height transition density of integral one inside $(0,2)$; its cumulative integral has slope at most one. Only this second-derivative bound is additional to the previously used cutoff properties.

For an input $h\in \mathrm{ran}{P}_{\epsilon }$ set

$$v=p_{N}Uh,Uh=\sqrt{\rho }h,f={\chi }_{R}v/\sqrt{\rho },t=R+2.$$

The Fourier multiplier is bounded by one and supported on $[-2N,2N]$. Plancherel gives $\Vert v^{(j)}{\Vert }_2\le (2N{)}^j\Vert h{\Vert }_{\nu }$ for $j=0,1,2$. Differentiating the physical $f$, rather than treating $Uh$ as physical-bandlimited, gives exactly

$$f^{\prime \prime }={\rho }^{-1/2}[{\chi }_R{v}^{\prime \prime }+2({\chi }_R^{\prime }-a{\chi }_R)v^{\prime }+({\chi }_R^{\prime \prime }-2a{\chi }_R^{\prime }+(a^2-a^{\prime }){\chi }_R)v].$$

Consequently (WA5) yields

$$\begin{array}{rl}\Vert f^{\prime \prime }W{\Vert }_2& \le B_{R,N}\Vert h{\Vert }_{\nu },\\ B_{R,N}& =A_W{e}^{4t}[4N^2+4N(1+L_{\rho }{e}^{4t})+C_{\chi }+2L_{\rho }{e}^{4t}+(L_{\rho }^2+H_{\rho })e^{8t}].\end{array}\text{\hspace{1em}(WA7)}$$

This bound cancels the inverse density against the comparison weight. It does not require a uniform unweighted bound for derivatives of ${\chi }_R/\sqrt{\rho }$.

Let ${\mathcal{P}}_{R,N,n}h=p_n(f)$. This is one bounded complex linear map into the even polynomials of degree at most $n+1$, and controls every unit input in the same spectral subspace. The spatial and Fourier errors of (FF1)–(FF2)/(WF8) are already at most $\sqrt{7}\tau /8$ and $\tau /3$, respectively, for the choices (FF7). Combining them with (WA6)–(WA7) gives

$$\Vert (I-{\mathcal{P}}_{R,N,n})P_{\epsilon }{\Vert }_{\nu \to \mathcal{F}}\le \sqrt{7}\tau /8+\tau /3+C_{\mathcal{F}}{C}_J{B}_{R,N}{a}_n/n.\text{\hspace{1em}(WA8)}$$

It therefore suffices to choose $n\ge 1$ with $C_{\mathcal{F}}{C}_J{B}_{R,N}{a}_n/n\le \tau /3$. This is an approximation of the whole spectral family, not a list of individual polynomial tests.

For an elementary degree estimate, the source’s defining MRS integral and $Q^{\prime }(r)\ge (\pi /2)e^{2r}$ for $r\ge 1$ give, when $a_n\ge 2$,

$$n=\frac{2}{\pi }{\int }_0^1\frac{a_{n}u{Q}^{\prime }(a_{n}u)}{\sqrt{1-u^2}}du\ge \frac{a_n}{8}{e}^{a_n}.$$

Only $1/2\le u\le 3/4$ is used for the last bound. Thus $a_n\le \log (8n)$ for every $n\ge 1$, including $a_n<2$. If $d=\max \{1,3C_{\mathcal{F}}{C}_J{B}_{R,N}/\tau \}$, the explicit choice

$$n=\lceil 4d\log (8d)\rceil \text{\hspace{1em}(WA9)}$$

has $n\ge d\log (8n)$ and pays the third term. For example, $4d\log (8d)\le n\le 5d\log (8d)$ and $\log (40d\log (8d))\le 4\log (8d)$ verify this inequality. The rank is at most $\lfloor (n+1)/2\rfloor +1$.

For fixed $\epsilon$, (FF7) has $N=O_{\epsilon }({\tau }^{-1}\log (e/\tau ))$ and $e^{4R}=O_{\epsilon }(\log (e/\tau {)}^2)$. Hence $B_{R,N}=O_{\epsilon }({\tau }^{-2}\log (e/\tau {)}^4)$, and (WA9) gives

$$\mathrm{rank}{\mathcal{P}}_{R,N,n}=O_{\epsilon }({\tau }^{-3}\log (e/\tau {)}^5).\text{\hspace{1em}(WA10)}$$

This asymptotic accuracy cost is smaller than the previously prescribed (FF8) cosine count. It does not give an evaluated practical rank: $C_J$ and the resulting constants are not numerically certified here. No conditioning bound or executable matrix certificate is supplied.

Remaining sign obligation

This is a conditional paper application of the unchanged theta model and the cited external analytic inequalities; it has not been Lean compiled and makes no research-priority claim. It reuses (WF2), the minimal-domain polynomial cutoff and the complete low-spectral Fourier map instead of rebuilding their prime or domain estimates.

With $\tau =\epsilon /8$, (WA8) provides the approximation input to the existing complete-window transfer. The retained matrix must still use the original full covariance and full jump form, with target $1/2-\epsilon /2$. The auxiliary $W^2$ mean only selects the polynomial’s constant; it replaces neither the original $\nu$ variance nor the target form. Positivity of those matrices and cofinal $\epsilon \downarrow 0$ certificates remain unproved. No all-degree PSD, RH, Robin, or FIB arithmetic advantage follows from this approximation bound.