Monomials form complete orthogonal systems of the Bergman spaces of the disc, the ball and the polydisc

The preceding monomial-integral lemma gives the three squared-norm formulas in the statement. Every value is positive and finite, so each monomial belongs to the corresponding Bergman space and can be normalized.

Let $\alpha \mathrm{\ne }\beta$ and choose $j<m$ with $d_0:={\beta }_j-{\alpha }_j\mathrm{\ne }0$. Set $d=\mathrm{\mid }{d}_0\mathrm{\mid }$ and $\eta =e^{i\pi \mathrm{/}d}$. The coordinate rotation $T(z{)}_j=\eta z_j$, fixing the other coordinates, maps $\mathrm{\Omega }$ and every $t\mathrm{\Omega }$ onto themselves and preserves Lebesgue measure by [F7]. For $g(z)=z^{\beta }\overline{z^{\alpha }}$, one has $g(Tz)={\eta }^{{\beta }_j}{\bar{\eta }}^{{\alpha }_j}g(z)={\eta }^{d_0}g(z)=-g(z)$: if $d_0>0$ the factor is ${\eta }^d=-1$, and if $d_0<0$ it is ${\eta }^{-d}=\overline{{\eta }^d}=-1$. Invariance [F8] gives $I:={\int }_{\mathrm{\Omega }}g={\int }_{\mathrm{\Omega }}g\circ T=-I$, so $I=0$ in $\mathbb{C}$. Integrability follows from [F2] and [F11]; therefore $⟨z^{\beta },z^{\alpha }⟩=0$. Hence distinct monomials are orthogonal, and the normalized family is orthonormal.

Fix $0<t<1$ and set $K_t=\overline{t\mathrm{\Omega }}$. By [F9], $K_t$ is compact and contained in $\mathrm{\Omega }$. For each $a\in K_t$, choose a positive polyradius $\rho$ with ${\bar{\mathrm{\Delta }}}_{\rho }(0)\subset \mathrm{\Omega }$ and $\mathrm{\mid }{a}_j\mathrm{\mid }<{\rho }_j$ for all $j<m$: if $\mathrm{\Omega }=\mathbb{D}$, take $\mathrm{\mid }a\mathrm{\mid }<\rho <1$; if $\mathrm{\Omega }={\mathbb{B}}^m$, take ${\rho }_j>\mathrm{\mid }{a}_j\mathrm{\mid }$ with ${\sum }_{j<m}{\rho }_j^2<1$; and if $\mathrm{\Omega }={\mathbb{D}}^m$, take $\mathrm{\mid }{a}_j\mathrm{\mid }<{\rho }_j<1$. These choices exist since $a\in \overline{t\mathrm{\Omega }}$ and $t<1$. Choose $s<\rho$ and $0<\theta <1$ with $\mathrm{\mid }{a}_j\mathrm{\mid }<\theta s_j$ for every $j<m$, possible because there are finitely many strict coordinate inequalities. Holomorphy gives the continuity and separate holomorphy required by [F4]. Define the box partial sums $S_N(z)={\sum }_{\alpha \in \{0,\dots ,N{\}}^m}{c}_{\alpha }{z}^{\alpha }$. By [F4], these sums converge uniformly on each closed polydisc strictly inside ${\mathrm{\Delta }}_{\rho }(0)$, including ${\bar{\mathrm{\Delta }}}_{\theta s}(0)$. If two admissible radii are used, restrict both expansions to a smaller common centered polydisc and apply [F5]; hence their coefficient families agree. The family of all such open polydiscs ${\mathrm{\Delta }}_{\theta s}(0)$ covers $K_t$, without selecting one for each point. By A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, this ambient open cover has a finite subcover; a common cutoff for its finitely many uniform convergences shows that these same box partial sums converge uniformly to $f$ on $K_t$.

Fix $0<t<1$ and a multi-index $\gamma$. The box partial sums $S_N$ from step 1.3 converge uniformly on $K_t$, so they are uniformly bounded there by a finite $M_t$. Since $\mathrm{\mid }{z}^{\gamma }\mathrm{\mid }\le 1$ on $\mathrm{\Omega }$ and $\lambda (t\mathrm{\Omega })<\mathrm{\infty }$ by [F13], the measurable functions ${\mathbf{1}}_{t\mathrm{\Omega }}{S}_N\overline{z^{\gamma }}$ are dominated by the integrable function $M_t{\mathbf{1}}_{t\mathrm{\Omega }}$; measurability follows from [F3], [F12] and [F16], and dominated convergence [F10] permits passing their integrals to the limit. For every $N\ge {\max }_j{\gamma }_j$, finite-sum linearity [F15] and the rotation argument of step 1.2 give ${\int }_{t\mathrm{\Omega }}{S}_N(z)\overline{z^{\gamma }}d{\lambda }_{2m}(z)=c_{\gamma }{\int }_{t\mathrm{\Omega }}\mathrm{\mid }{z}^{\gamma }{\mathrm{\mid }}^{2}d{\lambda }_{2m}(z)$, since every off-diagonal monomial moment vanishes on $t\mathrm{\Omega }$ by that same rotation argument. Taking the limit yields ${\int }_{t\mathrm{\Omega }}f(z)\overline{z^{\gamma }}d{\lambda }_{2m}(z)=c_{\gamma }{\int }_{t\mathrm{\Omega }}\mathrm{\mid }{z}^{\gamma }{\mathrm{\mid }}^{2}d{\lambda }_{2m}(z)$.

Suppose $f$ is orthogonal to every monomial. Let $t_n=1-1\mathrm{/}(n+2)$, so $t_n\uparrow 1$ and $t_n\mathrm{\Omega }$ increases to $\mathrm{\Omega }$. For each $\gamma$, the functions ${\mathbf{1}}_{t_n\mathrm{\Omega }}f\overline{z^{\gamma }}$ converge pointwise to $f\overline{z^{\gamma }}$ and are dominated by $\mathrm{\mid }f\overline{z^{\gamma }}\mathrm{\mid }$, which is integrable by [F11]; measurability follows from [F3], [F12] and [F16]. Also ${\mathbf{1}}_{t_n\mathrm{\Omega }}\mathrm{\mid }{z}^{\gamma }{\mathrm{\mid }}^2$ converges to $\mathrm{\mid }{z}^{\gamma }{\mathrm{\mid }}^2$ and is dominated by it, which is integrable by [F2]; its measurability follows by the same facts. Applying [F10] to both sequences and using step 2.1 gives $0=⟨f,z^{\gamma }⟩=c_{\gamma }\mathrm{\parallel }{z}^{\gamma }{\mathrm{\parallel }}_2^2$. The norm in [F2] is positive, so $c_{\gamma }=0$ for every $\gamma$.

Step 1.3 supplies the Taylor expansion on each $K_t$, and the dilates $t_n\mathrm{\Omega }$ exhaust $\mathrm{\Omega }$. Since all coefficients vanish by step 3.1, this expansion gives $f=0$ at every point of $\mathrm{\Omega }$. Thus the only element of $A^2(\mathrm{\Omega })$ orthogonal to every monomial is $0$.

Step 1.2 makes the normalized monomials an orthonormal family, and step 4.1 makes its orthogonal complement zero. By [F17] the Bergman space is Hilbert, so the zero-complement-to-completeness direction of [F14] shows that this family is complete. Conversely, if the family is complete, every vector orthogonal to its members is orthogonal to their dense linear span and hence, by Cauchy–Schwarz [F11], to itself; it must then be zero. Thus both directions of the stated equivalence hold, and the norm formulas and completeness establish all three asserted complete orthonormal systems.