$$, $$ {\mathbb {E}}\bigg[ \sup_{s\le t\wedge\tau_{n}}\|Y_{s}-Y_{0}\|^{2}\bigg] \le 2c_{2} {\mathbb {E}} \bigg[\int_{0}^{t\wedge\tau_{n}}\big( \|\sigma(Y_{s})\|^{2} + \|b(Y_{s})\|^{2}\big){\,\mathrm{d}} s \bigg] $$, $$\begin{aligned} {\mathbb {E}}\bigg[ \sup_{s\le t\wedge\tau_{n}}\!\|Y_{s}-Y_{0}\|^{2}\bigg] &\le2c_{2}\kappa{\mathbb {E}}\bigg[\int_{0}^{t\wedge\tau_{n}}( 1 + \|Y_{s}\| ^{2} ){\,\mathrm{d}} s \bigg] \\ &\le4c_{2}\kappa(1+{\mathbb {E}}[\|Y_{0}\|^{2}])t + 4c_{2}\kappa\! Given any set of polynomials \(S\), its zero set is the set. : Matrix Analysis. Also, = [1, 10, 9, 0, 0, 0] is also a degree 2 polynomial, since the zero coefficients at the end do not count. Suppose that you deposit $500 in a bank that offers an annual percentage rate of 6.0% compounded annually. Polynomials | Brilliant Math & Science Wiki Reading: Average Rate of Change. Thus \(\tau _{E}<\tau\) on \(\{\tau<\infty\}\), whence this set is empty. \(I\) \(\varepsilon>0\), By Ging-Jaeschke and Yor [26, Eq. This can be very useful for modeling and rendering objects, and for doing mathematical calculations on their edges and surfaces. To see that \(T\) is surjective, note that \({\mathcal {Y}}\) is spanned by elements of the form, with the \(k\)th component being nonzero. \(d\)-dimensional It process For \(i=j\), note that (I.1) can be written as, for some constants \(\alpha_{ij}\), \(\phi_{i}\) and vectors \(\psi _{(i)}\in{\mathbb {R}} ^{d}\) with \(\psi_{(i),i}=0\). Here the equality \(a\nabla p =hp\) on \(E\) was used in the last step. This is done throughout the proof. Differ. Applications of Taylor Polynomials - University of Texas at Austin By the above, we have \(a_{ij}(x)=h_{ij}(x)x_{j}\) for some \(h_{ij}\in{\mathrm{Pol}}_{1}(E)\). For instance, a polynomial equation can be used to figure the amount of interest that will accrue for an initial deposit amount in an investment or savings account at a given interest rate. Electron. Its formula yields, We first claim that \(L^{0}_{t}=0\) for \(t<\tau\). In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients that involves only the operations of addition, subtraction, multiplication, and. Anal. Thus if we can show that \(T\) is surjective, the rank-nullity theorem \(\dim(\ker T) + \dim(\mathrm{range } T) = \dim{\mathcal {X}} \) implies that \(\ker T\) is trivial. Camb. The following two examples show that the assumptions of LemmaA.1 are tight in the sense that the gap between (i) and (ii) cannot be closed. That is, \(\phi_{i}=\alpha_{ii}\). Since \(E_{Y}\) is closed this is only possible if \(\tau=\infty\). Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips, Not logged in is well defined and finite for all \(t\ge0\), with total variation process \(V\). J. Multivar. Let (eds.) and We have, where we recall that \(\rho\) is the radius of the open ball \(U\), and where the last inequality follows from the triangle inequality provided \(\|X_{0}-{\overline{x}}\|\le\rho/2\). Now consider any stopping time \(\rho\) such that \(Z_{\rho}=0\) on \(\{\rho <\infty\}\). Why learn how to use polynomials and rational expressions? Sci. $$, \(2 {\mathcal {G}}p({\overline{x}}) < (1-2\delta) h({\overline{x}})^{\top}\nabla p({\overline{x}})\), $$ 2 {\mathcal {G}}p \le\left(1-\delta\right) h^{\top}\nabla p \quad\text{and}\quad h^{\top}\nabla p >0 \qquad\text{on } E\cap U. 300, 463520 (1994), Delbaen, F., Shirakawa, H.: An interest rate model with upper and lower bounds. Discord. , The proof of Theorem4.4 follows along the lines of the proof of the YamadaWatanabe theorem that pathwise uniqueness implies uniqueness in law; see Rogers and Williams [42, TheoremV.17.1]. Let They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Let $$, $$ 0 = \frac{{\,\mathrm{d}}^{2}}{{\,\mathrm{d}} s^{2}} (q \circ\gamma_{i})(0) = \operatorname {Tr}\big( \nabla^{2} q(x) \gamma_{i}'(0) \gamma_{i}'(0)^{\top}\big) + \nabla q(x)^{\top}\gamma_{i}''(0), $$, \(S_{i}(x)^{\top}\nabla^{2} q(x) S_{i}(x) = -\nabla q(x)^{\top}\gamma_{i}'(0)\), $$ \operatorname{Tr}\Big(\big(\widehat{a}(x)- a(x)\big) \nabla^{2} q(x) \Big) = -\nabla q(x)^{\top}\sum_{i=1}^{d} \lambda_{i}(x)^{-}\gamma_{i}'(0) \qquad\text{for all } q\in{\mathcal {Q}}. This is not a nice function, but it can be approximated to a polynomial using Taylor series. \(L^{0}=0\), then Variation of constants lets us rewrite \(X_{t} = A_{t} + \mathrm{e} ^{-\beta(T-t)}Y_{t} \) with, where we write \(\sigma^{Y}_{t} = \mathrm{e}^{\beta(T- t)}\sigma(A_{t} + \mathrm{e}^{-\beta (T-t)}Y_{t} )\). where \(\widehat{b}_{Y}(y)=b_{Y}(y){\mathbf{1}}_{E_{Y}}(y)\) and \(\widehat{\sigma}_{Y}(y)=\sigma_{Y}(y){\mathbf{1}}_{E_{Y}}(y)\). \(\kappa>0\), and fix The use of polynomial diffusions in financial modeling goes back at least to the early 2000s. \(f\) Assume uniqueness in law holds for As when managing finances, from calculating the time value of money or equating the expenditure with income, it all involves using polynomials. For example: x 2 + 3x 2 = 4x 2, but x + x 2 cannot be written in a simpler form. This relies on(G1) and (A2), and occupies this section up to and including LemmaE.4. 3. \(\tau= \inf\{t \ge0: X_{t} \notin E_{0}\}>0\), and some Anal. The extended drift coefficient is now defined by \(\widehat{b} = b + c\), and the operator \(\widehat{\mathcal {G}}\) by, In view of (E.1), it satisfies \(\widehat{\mathcal {G}}f={\mathcal {G}}f\) on \(E\) and, on \(M\) for all \(q\in{\mathcal {Q}}\), as desired. Note that these quantities depend on\(x\) in general. The authors wish to thank Damien Ackerer, Peter Glynn, Kostas Kardaras, Guillermo Mantilla-Soler, Sergio Pulido, Mykhaylo Shkolnikov, Jordan Stoyanov and Josef Teichmann for useful comments and stimulating discussions. Finally, let \(\alpha\in{\mathbb {S}}^{n}\) be the matrix with elements \(\alpha_{ij}\) for \(i,j\in J\), let \(\varPsi\in{\mathbb {R}}^{m\times n}\) have columns \(\psi_{(j)}\), and \(\varPi \in{\mathbb {R}} ^{n\times n}\) columns \(\pi_{(j)}\). Thus we may find a smooth path \(\gamma_{i}:(-1,1)\to M\) such that \(\gamma _{i}(0)=x\) and \(\gamma_{i}'(0)=S_{i}(x)\). Activity: Graphing With Technology. Hence the following local existence result can be proved. Appl. It follows that the time-change \(\gamma_{u}=\inf\{ t\ge 0:A_{t}>u\}\) is continuous and strictly increasing on \([0,A_{\tau(U)})\). Taylor Polynomials. Swiss Finance Institute Research Paper No. 4. For any \(q\in{\mathcal {Q}}\), we have \(q=0\) on \(M\) by definition, whence, or equivalently, \(S_{i}(x)^{\top}\nabla^{2} q(x) S_{i}(x) = -\nabla q(x)^{\top}\gamma_{i}'(0)\). J. Financ. The generator polynomial will be called a CRC poly- To this end, define, We claim that \(V_{t}<\infty\) for all \(t\ge0\). Uses in health care : 1. Let \(Q^{i}({\mathrm{d}} z;w,y)\), \(i=1,2\), denote a regular conditional distribution of \(Z^{i}\) given \((W^{i},Y^{i})\). : The Classical Moment Problem and Some Related Questions in Analysis. }(x-a)^3+ \cdots.\] Taylor series are extremely powerful tools for approximating functions that can be difficult to compute . Registered nurses, health technologists and technicians, medical records and health information technicians, veterinary technologists and technicians all use algebra in their line of work. Sminaire de Probabilits XIX. Thus we obtain \(\beta_{i}+B_{ji} \ge0\) for all \(j\ne i\) and all \(i\), as required. Math. If \(i=j\), we get \(a_{jj}(x)=\alpha_{jj}x_{j}^{2}+x_{j}(\phi_{j}+\psi_{(j)}^{\top}x_{I} + \pi _{(j)}^{\top}x_{J})\) for some \(\alpha_{jj}\in{\mathbb {R}}\), \(\phi_{j}\in {\mathbb {R}}\), \(\psi _{(j)}\in{\mathbb {R}}^{m}\), \(\pi_{(j)}\in{\mathbb {R}}^{n}\) with \(\pi _{(j),j}=0\). Polynomials are easier to work with if you express them in their simplest form. [6, Chap. We need to show that \((Y^{1},Z^{1})\) and \((Y^{2},Z^{2})\) have the same law. are continuous processes, and The diffusion coefficients are defined by. Using that \(Z^{-}=0\) on \(\{\rho=\infty\}\) as well as dominated convergence, we obtain, Here \(Z_{\tau}\) is well defined on \(\{\rho<\infty\}\) since \(\tau <\infty\) on this set. list 3 uses of polynomials in healthcare. - Brainly.in Wiley, Hoboken (2004), Dunkl, C.F. Ann. Defining \(c(x)=a(x) - (1-x^{\top}Qx)\alpha\), this shows that \(c(x)Qx=0\) for all \(x\in{\mathbb {R}}^{d}\), that \(c(0)=0\), and that \(c(x)\) has no linear part. Springer, Berlin (1997), Penrose, R.: A generalized inverse for matrices. It involves polynomials that back interest accumulation out of future liquid transactions, with the aim of finding an equivalent liquid (present, cash, or in-hand) value. Two-term polynomials are binomials and one-term polynomials are monomials. Polynomials in one variable are algebraic expressions that consist of terms in the form axn a x n where n n is a non-negative ( i.e. Hence \(\beta_{j}> (B^{-}_{jI}){\mathbf{1}}\) for all \(j\in J\). An expression of the form ax n + bx n-1 +kcx n-2 + .+kx+ l, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree 'n' in variable x. After stopping we may assume that \(Z_{t}\), \(\int_{0}^{t}\mu_{s}{\,\mathrm{d}} s\) and \(\int _{0}^{t}\nu_{s}{\,\mathrm{d}} B_{s}\) are uniformly bounded. \end{aligned}$$, $$ \mathrm{Law}(Y^{1},Z^{1}) = \mathrm{Law}(Y,Z) = \mathrm{Law}(Y,Z') = \mathrm{Law}(Y^{2},Z^{2}), $$, $$ \|b_{Z}(y,z) - b_{Z}(y',z')\| + \| \sigma_{Z}(y,z) - \sigma_{Z}(y',z') \| \le \kappa\|z-z'\|. \(\mu\ge0\) 1123, pp. Assume for contradiction that \({\mathbb {P}} [\mu_{0}<0]>0\), and define \(\tau=\inf\{t\ge0:\mu_{t}\ge0\}\wedge1\). Start earning. 243, 163169 (1979), Article Anal. 18, 115144 (2014), Cherny, A.: On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. \(\varLambda\). We first deduce (i) from the condition \(a \nabla p=0\) on \(\{p=0\}\) for all \(p\in{\mathcal {P}}\) together with the positive semidefinite requirement of \(a(x)\). \(Y^{1}\), \(Y^{2}\) Thus, for some coefficients \(c_{q}\). for some constants \(\gamma_{ij}\) and polynomials \(h_{ij}\in{\mathrm {Pol}}_{1}(E)\) (using also that \(\deg a_{ij}\le2\)). Math. (ed.) Theorem3.3 is an immediate corollary of the following result. Similarly, \(\beta _{i}+B_{iI}x_{I}<0\) for all \(x_{I}\in[0,1]^{m}\) with \(x_{i}=1\), so that \(\beta_{i} + (B^{+}_{i,I\setminus\{i\}}){\mathbf{1}}+ B_{ii}< 0\). and the remaining entries zero. Ann. Inserting this into(F.1) yields, for \(t<\tau=\inf\{t: p(X_{t})=0\}\). PDF Polynomial Models in Finance - Universiteit van Amsterdam Polynomial can be used to calculate doses of medicine. Like actuaries, statisticians are also concerned with the data collection and analysis. Or one variable. Next, since \(\widehat{\mathcal {G}}p= {\mathcal {G}}p\) on \(E\), the hypothesis (A1) implies that \(\widehat{\mathcal {G}}p>0\) on a neighborhood \(U_{p}\) of \(E\cap\{ p=0\}\). earn yield. Since \(a(x)Qx=a(x)\nabla p(x)/2=0\) on \(\{p=0\}\), we have for any \(x\in\{p=0\}\) and \(\epsilon\in\{-1,1\} \) that, This implies \(L(x)Qx=0\) for all \(x\in\{p=0\}\), and thus, by scaling, for all \(x\in{\mathbb {R}}^{d}\). Specifically, let \(f\in {\mathrm{Pol}}_{2k}(E)\) be given by \(f(x)=1+\|x\|^{2k}\), and note that the polynomial property implies that there exists a constant \(C\) such that \(|{\mathcal {G}}f(x)| \le Cf(x)\) for all \(x\in E\). Let \(\gamma:(-1,1)\to M\) be any smooth curve in \(M\) with \(\gamma (0)=x_{0}\). 2023 Springer Nature Switzerland AG. We now focus on the converse direction and assume(A0)(A2) hold. 51, 361366 (1982), Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. arXiv:1411.6229, Lord, R., Koekkoek, R., van Dijk, D.: A comparison of biased simulation schemes for stochastic volatility models. All of them can be alternatively expressed by Rodrigues' formula, explicit form or by the recurrence law (Abramowitz and Stegun 1972 ). Polynomial regression - Wikipedia $$, $$ \|\widehat{a}(x)\|^{1/2} + \|\widehat{b}(x)\| \le\|a(x)\|^{1/2} + \| b(x)\| + 1 \le C(1+\|x\|),\qquad x\in E_{0}, $$, \({\mathrm{Pol}}_{2}({\mathbb {R}}^{d})\), \({\mathrm{Pol}} _{1}({\mathbb {R}}^{d})\), $$ 0 = \frac{{\,\mathrm{d}}}{{\,\mathrm{d}} s} (f \circ\gamma)(0) = \nabla f(x_{0})^{\top}\gamma'(0), $$, $$ \nabla f(x_{0})=\sum_{q\in{\mathcal {Q}}} c_{q} \nabla q(x_{0}) $$, $$ 0 \ge\frac{{\,\mathrm{d}}^{2}}{{\,\mathrm{d}} s^{2}} (f \circ\gamma)(0) = \operatorname {Tr}\big( \nabla^{2} f(x_{0}) \gamma'(0) \gamma'(0)^{\top}\big) + \nabla f(x_{0})^{\top}\gamma''(0). Springer, Berlin (1985), Berg, C., Christensen, J.P.R., Jensen, C.U. Let In particular, if \(i\in I\), then \(b_{i}(x)\) cannot depend on \(x_{J}\). To prove that \(X\) is non-explosive, let \(Z_{t}=1+\|X_{t}\|^{2}\) for \(t<\tau\), and observe that the linear growth condition(E.3) in conjunction with Its formula yields \(Z_{t} \le Z_{0} + C\int_{0}^{t} Z_{s}{\,\mathrm{d}} s + N_{t}\) for all \(t<\tau\), where \(C>0\) is a constant and \(N\) a local martingale on \([0,\tau)\). However, we have \(\deg {\mathcal {G}}p\le\deg p\) and \(\deg a\nabla p \le1+\deg p\), which yields \(\deg h\le1\). (x-a)+ \frac{f''(a)}{2!} In this appendix, we briefly review some well-known concepts and results from algebra and algebraic geometry. Taking \(p(x)=x_{i}\), \(i=1,\ldots,d\), we obtain \(a(x)\nabla p(x) = a(x) e_{i} = 0\) on \(\{x_{i}=0\}\). J. Probab. 5 uses of polynomial in daily life - Brainly.in \(\sigma:{\mathbb {R}}^{d}\to {\mathbb {R}}^{d\times d}\) Finance. International delivery, from runway to doorway. Econ. for all J. \end{aligned}$$, $$ {\mathbb {E}}\left[ Z^{-}_{\tau}{\boldsymbol{1}_{\{\rho< \infty\}}}\right] = {\mathbb {E}}\left[ - \int _{0}^{\tau}{\boldsymbol{1}_{\{Z_{s}\le0\}}}\mu_{s}{\,\mathrm{d}} s {\boldsymbol{1}_{\{\rho < \infty\}}}\right]. Step by Step: Finding the Answer (2 x + 4) (x + 4) - (2 x) (x) = 196 2 x + 8 x + 4 x + 16 - 2 . Now consider \(i,j\in J\). Ann. \({\mathbb {E}}[\|X_{0}\|^{2k}]<\infty \), there is a constant Similarly as before, symmetry of \(a(x)\) yields, so that for \(i\ne j\), \(h_{ij}\) has \(x_{i}\) as a factor. Replacing \(x\) by \(sx\), dividing by \(s\) and sending \(s\) to zero gives \(x_{i}\phi_{i} = \lim_{s\to0} s^{-1}\eta_{i} + ({\mathrm {H}}x)_{i}\), which forces \(\eta _{i}=0\), \({\mathrm {H}}_{ij}=0\) for \(j\ne i\) and \({\mathrm {H}}_{ii}=\phi _{i}\). Contemp. Example: x4 2x2 + x has three terms, but only one variable (x) Or two or more variables. \(z\ge0\). Noting that \(Z_{T}\) is positive, we obtain \({\mathbb {E}}[ \mathrm{e}^{\varepsilon' Z_{T}^{2}}]<\infty\). Let To see this, note that the set \(E {\cap} U^{c} {\cap} \{x:\|x\| {\le} n\}\) is compact and disjoint from \(\{ p=0\}\cap E\) for each \(n\). Exponents in the Real World | Passy's World of Mathematics \(Z\) \(Z\) We thank Mykhaylo Shkolnikov for suggesting a way to improve an earlier version of this result. A polynomial equation is a mathematical expression consisting of variables and coefficients that only involves addition, subtraction, multiplication and non-negative integer exponents of. Reading: Functions and Function Notation (part I) Reading: Functions and Function Notation (part II) Reading: Domain and Range. If \(i=j\ne k\), one sets. When On Earth Am I Ever Going to Use This? Polynomials In The - Forbes be the local time of Finally, LemmaA.1 also gives \(\int_{0}^{t}{\boldsymbol{1}_{\{p(X_{s})=0\} }}{\,\mathrm{d}} s=0\). Mark. \(X\) Its formula for \(Z_{t}=f(Y_{t})\) gives. : Abstract Algebra, 3rd edn. Furthermore, the drift vector is always of the form \(b(x)=\beta +Bx\), and a brief calculation using the expressions for \(a(x)\) and \(b(x)\) shows that the condition \({\mathcal {G}}p> 0\) on \(\{p=0\}\) is equivalent to(6.2). Google Scholar, Mayerhofer, E., Pfaffel, O., Stelzer, R.: On strong solutions for positive definite jump diffusions. Trinomial equations are equations with any three terms. 46, 406419 (2002), Article We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. Polynomials an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable (s). (eds.) EPFL and Swiss Finance Institute, Quartier UNIL-Dorigny, Extranef 218, 1015, Lausanne, Switzerland, Department of Mathematics, ETH Zurich, Rmistrasse 101, 8092, Zurich, Switzerland, You can also search for this author in These partial sums are (finite) polynomials and are easy to compute. Polynomials can be used to extract information about finite sequences much in the same way as generating functions can be used for infinite sequences. Polynomial brings multiple on-chain option protocols in a single venue, encouraging arbitrage and competitive pricing. $$, \(\tau_{E}=\inf\{t\colon X_{t}\notin E\}\le\tau\), \(\int_{0}^{t}{\boldsymbol{1}_{\{p(X_{s})=0\} }}{\,\mathrm{d}} s=0\), $$ \begin{aligned} \log& p(X_{t}) - \log p(X_{0}) \\ &= \int_{0}^{t} \left(\frac{{\mathcal {G}}p(X_{s})}{p(X_{s})} - \frac {1}{2}\frac {\nabla p^{\top}a \nabla p(X_{s})}{p(X_{s})^{2}}\right) {\,\mathrm{d}} s + \int_{0}^{t} \frac {\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s} \\ &= \int_{0}^{t} \frac{2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})}{2p(X_{s})} {\,\mathrm{d}} s + \int_{0}^{t} \frac{\nabla p^{\top}\sigma(X_{s})}{p(X_{s})}{\,\mathrm{d}} W_{s} \end{aligned} $$, $$ V_{t} = \int_{0}^{t} {\boldsymbol{1}_{\{X_{s}\notin U\}}} \frac{1}{p(X_{s})}|2 {\mathcal {G}}p(X_{s}) - h^{\top}\nabla p(X_{s})| {\,\mathrm{d}} s. $$, \(E {\cap} U^{c} {\cap} \{x:\|x\| {\le} n\}\), $$ \varepsilon_{n}=\min\{p(x):x\in E\cap U^{c}, \|x\|\le n\} $$, $$ V_{t\wedge\sigma_{n}} \le\frac{t}{2\varepsilon_{n}} \max_{\|x\|\le n} |2 {\mathcal {G}}p(x) - h^{\top}\nabla p(x)| < \infty. at level zero. Write \(a(x)=\alpha+ L(x) + A(x)\), where \(\alpha=a(0)\in{\mathbb {S}}^{d}_{+}\), \(L(x)\in{\mathbb {S}}^{d}\) is linear in\(x\), and \(A(x)\in{\mathbb {S}}^{d}\) is homogeneous of degree two in\(x\). Assessment of present value is used in loan calculations and company valuation. POLYNOMIALS USE IN PHYSICS AND MODELING Polynomials can also be used to model different situations, like in the stock market to see how prices will vary over time. A polynomial is a string of terms. \(c_{1},c_{2}>0\) The right-hand side is a nonnegative supermartingale on \([0,\tau)\), and we deduce \(\sup_{t<\tau}Z_{t}<\infty\) on \(\{\tau <\infty \}\), as required. Since polynomials include additive equations with more than one variable, even simple proportional relations, such as F=ma, qualify as polynomials. We now modify \(\log p(X)\) to turn it into a local submartingale. 13 Examples Of Algebra In Everyday Life - StudiousGuy Since \(\varepsilon>0\) was arbitrary, we get \(\nu_{0}=0\) as desired. Share Cite Follow answered Oct 22, 2012 at 1:38 ILoveMath 10.3k 8 47 110 For each \(m\), let \(\tau_{m}\) be the first exit time of \(X\) from the ball \(\{x\in E:\|x\|< m\}\). \(f\in C^{\infty}({\mathbb {R}}^{d})\) Then, for all \(t<\tau\). . 29, 483493 (1976), Ethier, S.N., Kurtz, T.G. The proof of relies on the following two lemmas. V.26]. To prove(G2), it suffices by Lemma5.5 to prove for each\(i\) that the ideal \((x_{i}, 1-{\mathbf {1}}^{\top}x)\) is prime and has dimension \(d-2\). $$, \(f,g\in {\mathrm{Pol}}({\mathbb {R}}^{d})\), https://doi.org/10.1007/s00780-016-0304-4, http://e-collection.library.ethz.ch/eserv/eth:4629/eth-4629-02.pdf. Google Scholar, Stoyanov, J.: Krein condition in probabilistic moment problems.
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