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A compact convex set $K\subseteq {\mathbb{R}}^n$ is a compact set that is also a convex set. If $K$ is nonempty, its support function $H_K(\eta )={\max }_{x\in K}x\cdot \eta$ is finite in every direction. For such a nonempty $K$, every exterior point $x_0$ admits a strictly separating hyperplane: minimize $|x_0-y|$ over $y\in K$, and put $\omega =x_0-y$. Convexity and differentiation along the segment from $y$ to any $z\in K$ give $\omega \cdot (z-y)\le 0$, while $\omega \cdot (x_0-y)=|\omega {|}^2>0$. This is the separation used to recover spatial support from Fourier growth.

An extreme point of a convex set is a point that cannot be made by genuinely mixing two different points of the set. For a convex set $C$ in a real vector space, the condition on $p\in C$ isThus $p$ lies on no open line segment between two distinct points of $C$. It is enough to exclude midpoint representations: if $p$ lies inside a line segment, two nearby points on opposite sides have midpoint $p$.^[ref]

The square $[-1,1{]}^2$ has just four extreme points, at $(\pm 1,\pm 1)$. In a convex combination giving a corner, each coordinate is already at its largest or smallest allowed value, forcing both input points to agree with that corner. In contrast, the boundary point $(1,0)$ is not extreme:It is the midpoint of two other points on the same edge. An interior point also admits a small line segment through it.

The unit disk $D=\{z\in {\mathbb{R}}^2:\Vert z{\Vert }_2\le 1\}$ has a different shape: every point on its unit circle is extreme. For a unit vector $p$, the Cauchy-Schwarz inequality givesand equality is possible only at $z=p$. If a convex combination equals $p$, applying this linear functional forces both ingredients to attain that equality. Both must therefore be $p$. Extremality need not look like a sharp corner.

These points help describe all allowable mixtures. Every $(x,y)\in [-1,1{]}^2$ is a convex combination of its four corners, with the weight on $(a,b)\in \{-1,1{\}}^2$ equal to $(1+ax)(1+by)/4$. For the unit disk, write $z=rp$, with $0\le r\le 1$ and $p$ a unit vector; at zero choose any $p$. ThenTwo extreme points suffice for that mixture. The Krein-Milman theorem gives a much broader reconstruction: a nonempty compact convex set in a locally convex space whose topology is Hausdorff is the closed convex hull of its extreme points.

Mark Krein and David Milman published their reconstruction theorem in 1940. Milman's colleagues describe that result and its subsequent use as a method of representing convex sets through their extreme points.^[ref]

A nonempty compact convex set $K$ in a finite-dimensional space and an upper hemicontinuous set-valued mapping $F:K\rightrightarrows K$ with nonempty closed convex values admit $x\in F(x)$. A closed graph suffices when the whole output space is compact.

The essential two-person bargaining domain consists of a compact convex set of feasible utility vectors $F$, a disagreement point $d\in F$, and at least one feasible vector strictly exceeding $d$ coordinatewise. The Nash bargaining solution selects a jointly feasible improvement using a rule characterized by Pareto efficiency, bargaining symmetry, positive affine invariance in bargaining and bargaining independence of irrelevant alternatives.

For a compact convex set of feasible payoff vectors $F$ and a specified disagreement point $d$, the negotiation set is its individually rational Pareto frontier. It is the set of efficient agreements each player prefers at least weakly to disagreement. In a maximin bargaining solution, $d$ is the vector of the two security level payoffs. This usage is distinct from objection-and-counterobjection bargaining sets in coalitional-game theory.

In a two-person Nash bargaining problem, an agreement specifies a jointly feasible pair of payoffs $(u,v)$, measured in each player's own utility function. For a finite game, binding agreements can randomize jointly among action pairs, so the feasible payoff region $F$ is the convex hull of the pure-action payoff. This permits correlated joint actions; it is generally larger than the payoff region generated by independent mixed strategies. A disagreement point $d=(d_1,d_2)$ specifies what each player receives if negotiations fail. Assume a compact convex set of feasible agreements and at least one agreement strictly improving both disagreement payoffs.

A vector $(u^{\prime },v^{\prime })$ jointly dominates $(u,v)$ if $u^{\prime }\ge u$, $v^{\prime }\ge v$, and at least one inequality is strict. This is joint dominance of payoff vectors; strict improvement for both is unnecessary. A Pareto optimal agreement is one that has no feasible joint dominator. A jointly dominated agreement wastes a possible gain, but Pareto efficiency alone does not settle how gains should be divided. The negotiation set in two-person bargaining isIts individual-rationality constraints keep either player from accepting less than disagreement; its Pareto efficiency constraint rules out wasted opportunities. This negotiation set is distinct from objection-and-counterobjection bargaining sets for coalitional games.

The Nash arbitration procedure selects the Nash bargaining solution by maximizing gains relative to disagreement:The objective function is the Nash product. Essentiality makes its maximum positive; compactness gives attainment. On positive gains, its logarithm is $\log (u-d_1)+\log (v-d_2)$, whose Hessian is diagonal and negative definite. Thus strict concavity on the feasible convex set makes the maximizing payoff vector unique. Positive gain in either coordinate increases the product, so the solution is Pareto optimal.

The procedure respects feasibility, bargaining individual rationality, Pareto efficiency, bargaining symmetry, positive affine invariance in bargaining, and bargaining independence of irrelevant alternatives. In particular, replacing each player's utility by a separate positive affine rescaling multiplies the product by a positive constant; it does not compare one player's raw utility scale to the other's. Deleting other feasible agreements cannot change a unique maximizer that remains feasible. Symmetry under interchange of the players gives equal selected coordinates when the bargaining problem itself is symmetric.

These properties also explain why the product rule is distinguished. Normalize a positive-gain maximizer to $(1,1)$ and disagreement to $(0,0)$ by positive affine maps. The first-order inequality along a segment from $(1,1)$ to any feasible $(a,b)$ gives $a+b\le 2$. A sufficiently large symmetric triangle, bounded below in each coordinate and above by $a+b=2$, contains the normalized feasible region. Symmetry and Pareto efficiency select $(1,1)$ in that triangle. Independence of irrelevant alternatives then selects the same point in the original normalized region, and affine invariance transforms it back. This is the supporting triangle for Nash bargaining argument; the assumptions of convexity and a strict joint improvement are important. Without strict improvement, a zero product need not select a unique agreement.

For a finite game, a natural disagreement convention is each player's security level payoff:where $A,B$ are the two payoff matrices and $p,q$ are mixed strategies. Each player can independently guarantee at least its own security level. The maximin bargaining solution applies the Nash product rule using this $d$. In this convention, “maximin” determines the disagreement payoffs; it does not replace the arbitration objective function by $\min (u,v)$, which would depend improperly on interpersonal utility scales. The resulting agreement can strictly benefit both players relative to their guarantees, yet need not benefit both relative to a particular noncooperative Nash equilibrium.

A Nash bargaining problem consists of a compact convex set $S$ of feasible two-player payoff vectors and a disagreement point $d\in S$. If bargaining fails, the players receive $d$; agreement selects a feasible vector. The convex set allows jointly agreed correlated payoff lotteries over outcomes. Assume essentiality: some feasible vector strictly improves both coordinates of $d$. The admissible answers satisfy bargaining individual rationality, $u_i\ge d_i$.

The four characterizing axioms are Pareto efficiency (no feasible vector weakly improves both chosen payoffs and strictly improves one), bargaining symmetry (interchanging identical player roles leaves the answer unchanged), positive affine invariance in bargaining (independently replacing utilities by $a_i{u}_i+b_i$, $a_i>0$, transforms the answer by the same maps), and bargaining independence of irrelevant alternatives (restricting to an admissible smaller feasible set that still contains the chosen vector leaves the choice unchanged). The last axiom keeps the disagreement point fixed.

The arbitration procedure maximizes the Nash product over individually rational vectors:A maximizer exists by compactness and has positive gains by essentiality. It is unique, since the logarithm of the product is a strictly concave function of the two gains on their positive domain. The product is strictly increasing in either positive gain, giving Pareto efficiency. A positive affine utility change multiplies the product by $a_1{a}_2$, giving positive affine invariance in bargaining. Uniqueness gives bargaining symmetry and bargaining independence of irrelevant alternatives.

For completeness, these axioms also force the procedure. Normalize the product-maximizing vector to $(1,1)$ and $d$ to zero using positive affine invariance in bargaining. For any normalized feasible vector $z$, the directional derivative of the product along the feasible segment from $(1,1)$ to $z$ is $z_1+z_2-2$. The initial part of that segment has positive gains, so optimality implies $z_1+z_2\le 2$. Choose $M$ sufficiently large that the transformed feasible set is contained in the symmetric triangleOn this triangle, bargaining symmetry requires equal coordinates and Pareto efficiency then forces $(1,1)$. By bargaining independence of irrelevant alternatives, restricting from $T_M$ to the original normalized set still selects $(1,1)$. Undoing the normalization proves the Nash bargaining solution characterization. This is the supporting triangle for Nash bargaining argument.

For the numerical game, the maximin disagreement point is the vector of the players' separate security level payoffs. If player I uses its first action with probability $p$, its worst expected payoff isThe two lines meet at $p=1/4$; before that point the increasing second line is smaller and afterwards the decreasing first line is smaller. Hence $d_1=7/2$. If player II uses its first action with probability $q$, its worst payoff ismaximized at $q=1$, giving $d_2=4$. Thus $d=(7/2,4)$. These are separate worst-case guarantees; they need not equal the actual payoffs when both security strategies are used together.

The cooperative feasible set is the convex hull of the four outcome vectors. The outcomes $(2,4)$ and $(2,3)$ are dominated by $C=(4,5)$. Its Pareto frontier is the segment joining $C$ to $B=(8,2)$, with equation $y=8-3x/4$. Any maximizer of the positive Nash product is on this frontier. Bargaining individual rationality restricts its relevant portion to $4\le x\le 16/3$. There we maximizeThe stationary point lies strictly inside that interval and is the unique maximum. Therefore the Nash bargaining solution isThe gains are $11/12$ and $11/16$, with Nash product $121/192$. The arbitration can implement the vector by the agreed correlated payoff lottery assigning probability $5/48$ to outcome $B$ and $43/48$ to outcome $C$. Indeed $4+4(5/48)=53/12$ and $5-3(5/48)=75/16$. This is a correlated payoff lottery; independent mixed strategies are not required to implement the cooperative agreement.

For $v\in Z$, write $M(v)={\max }_i{v}_i$. Maximizing a linear functional on the price simplex givesThus $g(v)$ is a nonempty closed convex face of the compact price simplex. Its graph of a set-valued mapping is closed: if $v_n\to v$, $p_n\to p$ and $p_n\in g(v_n)$, then, for every fixed $q\in P$, the inequalities $p_n\cdot v_n\ge q\cdot v_n$ pass to $p\cdot v\ge q\cdot v$.

Each value $z(p)$ is closed in $Z$ by the closed-graph hypothesis, hence compact; it is nonempty and convex by assumption. Therefore $F(p,v)=g(v)\times z(p)$ has nonempty closed convex values in the nonempty compact convex set $K=P\times Z$.

Its graph of a set-valued mapping is also closed. If $(p_n,v_n)\to (p,v)$ and $(q_n,w_n)\to (q,w)$ with $q_n\in g(v_n)$ and $w_n\in z(p_n)$, the two closed-graph properties give $q\in g(v)$ and $w\in z(p)$. Finally, this closed graph implies upper hemicontinuity because the whole output space $K$ is compact. If an open set containing $F(p,v)$ failed to contain the nearby values, select outputs outside it along a convergent input sequence. A subsequence of those outputs converges in $K$; closedness puts its limit in $F(p,v)$, whereas closedness of the open set's complement puts it outside. This contradiction proves upper hemicontinuity.

Consequently all the hypotheses of the Kakutani fixed-point theorem hold: the domain is nonempty, compact and convex, and the set-valued mapping maps it into itself with nonempty closed convex values and upper hemicontinuity.

Let $f(x)=-{\sum }_i\log ({\alpha }_i+x_i)$. The feasible probability simplex is a nonempty compact convex set, so the extreme value theorem guarantees a minimizer. Its Hessian matrix is diagonal with positive entries $({\alpha }_i+x_i{)}^{-2}$, making $f$ a strictly convex function. Hence the minimizer is unique.

Use the Lagrange multipliers $\lambda$ for the sum constraint and ${\nu }_i\ge 0$ for $-x_i\le 0$. The Lagrangian isThe KKT conditions areA strictly positive feasible allocation exists, so the Slater condition holds. These conditions are necessary and sufficient for this convex optimization. At least one coordinate is positive, giving $\lambda >0$. Put $\tau =1/\lambda$. For a positive coordinate, complementary slackness gives ${\alpha }_i+x_i=\tau$. At a zero coordinate, stationarity gives ${\nu }_i=1/\tau -1/{\alpha }_i\ge 0$, or ${\alpha }_i\ge \tau$. ThusHere $(u{)}_{+}=\max (u,0)$. This logarithmic water filling raises the smaller baseline values to a common level. The scalar left side is a continuous function and strictly increasing above ${\min }_i{\alpha }_i$, starts at zero, and tends to infinity, so exactly one $\tau$ solves it. One can find it by interval bisection, or sort the baseline values increasingly and find an index $k$ for whichusing ${\alpha }_{(n+1)}=+\infty$. Equality at the next baseline simply gives a zero allocation. Sorting and a running sum implement the water-filling algorithm in $O(n\log n)$ time.

Let $P$ be the row payoff matrix and $Q=P^T$. For $x$ in the probability simplex $\Delta$, write $u(x)=x^{T}Px$ and define the symmetric Nash gain mapThe map is a continuous function, has nonnegative coordinates, and sums to one. The Brouwer fixed-point theorem states that every continuous self-map of a nonempty finite-dimensional compact convex set has a fixed point. Apply it to $T:\Delta \to \Delta$, and let $x=T(x)$.

Set $G={\sum }_i{g}_i(x)$. The fixed-point equation gives $g_i(x)=G{x}_i$. If $G>0$, every positive $x_i$ has strictly positive gain, so $(Px{)}_i-u(x)=g_i(x)=G{x}_i$. Buta contradiction. Thus $G=0$ and every pure payoff $(Px{)}_i$ is at most $u(x)$. Since their $x$-weighted average equals $u(x)$, every supported action attains that maximum. Consequently $x$ is a best response to itself.

The column player's payoff vector against $x$ is $Q^{T}x=Px$, so exactly the same inequalities establish its best response. ThereforeThis supplies the whole fixed-point construction and fixed-point-to-equilibrium argument, rather than assuming Nash's theorem as a black box.

Use the usual essential two-person Nash bargaining problem: $F\subset {\mathbb{R}}^2$ is a compact convex set, $d\in F$ is the disagreement point, and some $u\in F$ satisfies $u_i>d_i$ for both players. The Nash bargaining solution isThe positive maximum exists by compactness and essentiality. On positive gains, maximizing this Nash product is equivalent to maximizing $\log (u_1-d_1)+\log (u_2-d_2)$, a strictly concave function. Convexity then gives a unique maximizer. The essentiality and compactness hypotheses matter: for example, with $F=[0,1]\times \{0\}$ and $d=0$, the product is zero everywhere and its argmax alone is not a single-valued definition.

The rule satisfies all four axioms. Pareto efficiency: a feasible vector dominating the chosen vector with at least one strict improvement would increase its positive Nash product. Bargaining symmetry: if the problem is unchanged by swapping players, uniqueness makes the answer unchanged, so the two payoffs agree. Positive affine invariance in bargaining: for $v_i=a_i{u}_i+b_i$ with $a_i>0$, gains transform to $a_i(u_i-d_i)$ and the product is multiplied by the positive constant $a_1{a}_2$, preserving its maximizer. Bargaining independence of irrelevant alternatives: if $G\subseteq F$ is another admissible feasible set containing the chosen vector and the same disagreement point, that vector remains the unique product maximizer over $G$.

To prove characterization, let $f$ be any feasible single-valued rule satisfying these axioms, and let $u^{\ast }=N(F,d)$. Normalize payoffs by the positive affine transformationThe transformed set $H$ has disagreement point zero and product maximizer $e=(1,1)$. For any $z\in H$, convexity puts $e+t(z-e)$ in $H$; for sufficiently small $t>0$ both gains remain positive. The one-sided derivative of the product at its maximum is therefore nonpositive:Thus $H\subseteq \{z:z_1+z_2\le 2\}$. By compactness choose $M\ge 0$ so every coordinate of every $z\in H$ is at least $-M$. The supporting triangle for Nash bargaining isIt contains $H$, is compact, convex, symmetric and essential, and contains disagreement zero. Symmetry forces $f(T_M,0)$ onto the diagonal; Pareto efficiency then forces it to be $e=(1,1)$. Since $e\in H\subseteq T_M$, bargaining independence of irrelevant alternatives gives $f(H,0)=e$. Undoing the normalization by positive affine invariance in bargaining gives $f(F,d)=u^{\ast }$. Hence the four axioms uniquely characterize the Nash bargaining solution on this domain.

Use the Fourier transform convention $\hat{\phi }(\xi )=\int e^{-ix\cdot \xi }\phi (x)dx$, with inverse factor $(2\pi {)}^{-n}$. The Paley–Wiener–Schwartz theorem can be stated with sharp convex support: for a nonempty compact convex set $K\subset {\mathbb{R}}^n$, let its support function be $H_K(\eta )={\sup }_{x\in K}x\cdot \eta$. Then $F$ is the Fourier transform of a unique distribution supported in $K$ if and only if it is entire andfor some $C$ and nonnegative integer $M$. For $K={\overline{B}}_R$, this is the familiar bound $C(1+|z|{)}^M{e}^{R|\mathrm{Im}z|}$. Allowing some $R$ characterizes all compactly supported distributions. Here the extension of the Fourier transform is $F(z)=⟨u,e^{-ix\cdot z}⟩$, interpreted with a cutoff equal to one near the support of a distribution.

First suppose $\mathrm{supp}u\subset K$. Fix one smooth cutoff function equal to one near $K$. Pairing the resulting compactly supported exponential with $u$ shows that $F$ is an entire function: differentiation with respect to $z_j$ inserts $-i{x}_j$, and the power series converges in the test-function seminorms uniformly on compact subsets of ${\mathbb{C}}^n$.

To keep the exponential type exactly $H_K$, rather than that of a fixed larger neighborhood, use a shrinking-cutoff exponential-type estimate. There are cutoffs ${\chi }_{\epsilon }$ equal to one on $K+\epsilon B_1$, supported in $K+3\epsilon {\overline{B}}_1$, and satisfying $|{\partial }^{\alpha }{\chi }_{\epsilon }|\le C_{\alpha }{\epsilon }^{-|\alpha |}$ for $0<\epsilon \le 1$. One construction convolves the indicator of $K+2\epsilon B_1$ with a unit-mass mollifier supported in $\epsilon B_1$. Continuity of $u$ on a fixed compact neighborhood gives a finite order of a distribution $q$ there. By the product rule,Taking $\epsilon =(1+|z|{)}^{-1}$ proves the required bound with $M=2q$, since the extra exponential factor is at most $e^3$. The value of the pairing is independent of the chosen cutoff because all cutoffs agree near $\mathrm{supp}u$. Thus the forward direction has the exact asserted support function, without an unproved estimate on derivatives restricted only to $K$.

Conversely, suppose the entire $F$ has the stated bound. Its restriction to real frequency has polynomial growth, so define a tempered distribution $u$ byThe Schwartz space decay makes this integral absolutely convergent and continuous; by Fourier inversion, $\hat{u}=F$ on real frequency. It remains to prove the support assertion by mollifier regularization for contour recovery of support.

Choose a nonnegative unit-mass mollifier $\rho$ supported in $B_1$, and setOn real frequency, $F_{\epsilon }$ decays faster than any polynomial after enough applications of integration by parts to $\hat{\rho }$; hence $h_{\epsilon }$ is smooth, by differentiation under the integral sign. More generally, for every integer $L$ there is $C_{\epsilon ,L}$ such thatTo obtain this estimate, write the transform of ${\rho }_{\epsilon }$ as the real-frequency transform of $e^{x\cdot \eta }{\rho }_{\epsilon }(x)$ and integrate by parts; each derivative introduces at most one factor of $|\eta |$.

Fix a unit vector $v$ and take $L>M+n+1$. A contour-shift proof of the Paley–Wiener–Schwartz theorem moves the inverse-transform contour to ${\mathbb{R}}^n+itv$:Here is a justification of the shift. Rotate coordinates so $v$ is the first coordinate direction, apply the Cauchy integral theorem on a rectangle in the first complex variable, and integrate over the other real variables. For fixed $t$, the vertical sides at real part $\pm A$ have an integrated bound proportional to $A^{M-L+n-1}$, and hence vanish as $A\to \infty$. The same decay bounds give absolute convergence on the horizontal sides. No contour shift of an unregularized polynomially growing integral is needed.

Positive homogeneity of the support function now givesIf $x\notin K+\epsilon {\overline{B}}_1$, the Hahn-Banach separation theorem supplies a unit vector $v$ with $x\cdot v>H_K(v)+\epsilon$. Letting $t\to \infty$ proves $h_{\epsilon }(x)=0$. Therefore $h_{\epsilon }\in C_c^{\infty }$ and $\mathrm{supp}{h}_{\epsilon }\subset K+\epsilon {\overline{B}}_1$.

Since $\hat{\rho }(\epsilon \xi )\to 1$ and its modulus is at most one on real frequency, the dominated convergence theorem in the formula for $⟨u,\phi ⟩$ gives $h_{\epsilon }\to u$ as tempered distributions, and thus as distributions. A test function supported outside $K$ has positive distance from $K$, so its pairing with $h_{\epsilon }$ is zero for all sufficiently small $\epsilon$. This proves $\mathrm{supp}u\subset K$. The Fourier transform of a compactly supported distribution constructed in the forward direction equals $F$ on ${\mathbb{R}}^n$; applying the one-variable identity theorem successively in each coordinate extends the equality to ${\mathbb{C}}^n$. Injectivity of the Fourier transform of a tempered distribution proves uniqueness and completes both directions.

For the independence application, put $g_m(z)=e^{iz\cdot y_m}{f}_m(z)$ and $r_m=1/(m+1)$. The ball version of the Paley–Wiener–Schwartz theorem supplies a nonzero distribution $v_m$ supported in ${\overline{B}}_{r_m}(0)$ with ${\hat{v}}_m=g_m$. By the Translation property of the Fourier transform,where $⟨{\tau }_{y}v,\phi ⟩=⟨v,\phi (\cdot +y)⟩$. Distinct integer points are at least distance one apart. For distinct positive indices $j,k$, the largest possible radius sum is $1/2+1/3=5/6<1$. Thus these closed balls, and hence the distribution supports, are pairwise disjoint.

If ${\sum }_{m=1}^N{c}_m{f}_m=0$, injectivity of the Fourier transform gives ${\sum }_m{c}_m{w}_m=0$. For each $j$, choose a smooth cutoff function equal to one near its ball and zero near all other balls. Multiplying the distributional identity by this cutoff isolates $c_j{w}_j=0$. Since $f_j$ is not identically zero, $w_j\ne 0$, so $c_j=0$. Consequently $f_1,\dots ,f_N$ are linearly independent over $\mathbb{C}$. This Fourier independence from disjoint distribution supports uses the quantitative exponential types to establish support separation.

Use the Fourier transform convention $\hat{\phi }(\xi )=\int e^{-ix\cdot \xi }\phi (x)dx$ and $D_j=-i{\partial }_j$, so $\widehat{P(D)u}=P(\xi )\hat{u}$. All distribution pairings use complex bilinearity, and the formal transpose is $P(-D)$, without complex conjugation of the coefficients. If $D_j$ instead denotes ${\partial }_j$, the Fourier multiplier is $P(i\xi )$; the argument below is unchanged after this substitution.

For a nonempty compact convex set $K\subset {\mathbb{R}}^n$, let $H_K(\eta )={\sup }_{x\in K}x\cdot \eta$ be its support function. The Paley–Wiener–Schwartz theorem says that the Fourier transform is a bijection between distributions supported in $K$ and entire functions $F$ on ${\mathbb{C}}^n$ for whichfor some $C>0$ and nonnegative integer $N$. In particular, support in ${\overline{B}}_R$ is equivalent to the bound with $e^{R|\mathrm{Im}\zeta |}$. Convexity is essential in the precise support formulation: the support function of a set equals that of its convex hull.

Suppose first that $u$ is a compactly supported distribution with support in $K$. Define $F(\zeta )=⟨u,e^{-ix\cdot \zeta }⟩$, inserting any cutoff function equal to one near $K$ when viewing $u$ on test functions. A fixed cutoff and the finite order of a distribution estimate permit differentiation of the pairing in each complex variable, giving ${\partial }_{\zeta }^{\alpha }F=⟨u,(-ix{)}^{\alpha }{e}^{-ix\cdot \zeta }⟩$. The exponential function's power series converges with every required derivative on a fixed compact set; hence $F$ is an entire function. On real frequencies it agrees with the Fourier transform of a compactly supported distribution.

The exact exponential indicator needs a shrinking cutoff function, rather than one fixed outside $K$. Choose ${\chi }_{\epsilon }=1$ near $K$, with support in $K+B_{2\epsilon }$ and $\Vert {\partial }^{\alpha }{\chi }_{\epsilon }{\Vert }_{\infty }\le C_{\alpha }{\epsilon }^{-|\alpha |}$ for $0<\epsilon \le 1$. Such a cutoff is obtained by convolving the indicator of $K+B_{\epsilon }$ with a mollifier supported in $B_{\epsilon /2}$. On one fixed ball containing all these supports, the finite order of a distribution bound isThe Leibniz rule applied to $\theta ={\chi }_{\epsilon }{e}^{-ix\cdot \zeta }$ therefore givesTaking $\epsilon =(1+|\zeta |{)}^{-1}$ proves the required bound, with $N=2m$ and a fixed additional factor at most $e^2$.

Conversely, suppose $F$ is an entire function with the displayed bound. Its real restriction has polynomial growth, so its inverse Fourier transform defines a tempered distribution byRapid decay of the Schwartz function $\hat{\phi }$ makes this integral absolutely convergent and continuous in the Schwartz space topology.

To determine the support of a distribution, take a test function supported in a half-space $x\cdot \omega \ge H_K(\omega )+\delta$, where $|\omega |=1$ and $\delta >0$. The contour-shift proof of the Paley–Wiener–Schwartz theorem gives, for each $t\ge 0$,Here is the estimate that justifies both the shift and its limiting use. Since $\hat{\phi }(-\xi -it\omega )=\int e^{ix\cdot \xi }{e}^{-tx\cdot \omega }\phi (x)dx$, repeated integration by parts with $1-{\Delta }_x$ yieldsFor fixed $t$, the same estimate is uniform over the imaginary strip from $0$ to $t\omega$. Rotate coordinates so $\omega$ is the first coordinate direction, apply the Cauchy integral theorem on truncated rectangles, and let their real edges tend to infinity. If $2L>N+n$, the boundary integrals vanish and the horizontal integrals converge absolutely; thus the claimed shift is valid. Using $H_K(t\omega )=t{H}_K(\omega )$ now bounds the pairing bywhich tends to zero. A separating hyperplane exists at every point outside the compact convex set $K$. A finite partition of unity on the support of any exterior test function reduces it to such half-spaces, so $\mathrm{supp}u\subseteq K$. Fourier inversion gives uniqueness; the entire transform of this $u$ agrees with $F$ on ${\mathbb{R}}^n$ and hence everywhere, by applying the one-variable identity theorem successively in each coordinate. This completes both directions.

Now apply this theorem to compact support solvability for a constant-coefficient ordinary differential equation. If $P$ is a nonzero constant, its kernel is zero and $u=v/P$ is the unique solution. Otherwise writeThe solution space of $P(-D)\phi =0$ isIndeed these exponentials solve the equation, their initial derivative vectors have a nonzero Vandermonde determinant, and uniqueness for the order-$m$ ordinary differential equation makes them a basis. Pairing $P(D)u=v$ with any such solution proves necessity:

For sufficiency, choose $R$ with $\mathrm{supp}v\subseteq [-R,R]$ and let $V(\zeta )=⟨v,e^{-ix\zeta }⟩$. The assumed annihilation is precisely $V({\alpha }_j)=0$ for every $j$. Since all roots of a polynomial are simple, $U=V/P$ has removable singularities at every root and is an entire function. Outside one disk, $|P(\zeta )|\ge c(1+|\zeta |{)}^m$. Inside that disk, the extended $U$ is bounded. The polynomial division preservation of exponential type therefore givesThe Paley–Wiener–Schwartz theorem produces a compactly supported distribution $u$ with transform $U$, supported in the same interval. Then $\widehat{P(D)u}=PU=V$, so Fourier inversion gives $P(D)u=v$. ThusThe compactly supported solution is unique: $P\hat{u}=0$ forces the entire transform to vanish off the finitely many roots and hence everywhere. With repeated roots, the corresponding condition would also require derivatives $V^{(k)}(\alpha )=0$, or pairing with $x^k{e}^{-i\alpha x}$, up to one less than each multiplicity.

Let a distribution have support in a compact convex set $K$. Fixed neighborhood cutoffs initially give an exponential bound larger than the support function $H_K$. Cutoffs of width $\epsilon$ cost powers of ${\epsilon }^{-1}$ in derivative estimates but enlarge the exponential by only $e^{C\epsilon |\mathrm{Im}z|}$. Choosing $\epsilon =(1+|z|{)}^{-1}$ absorbs the cutoff cost into a polynomial and retains the exact exponential $e^{H_K(\mathrm{Im}z)}$.