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Let

H(D)

denote the space of functions holomorphic on the disc

D=\{ z\colon |z|<1 \}

, endowed with the topology of uniform convergence on each compact subset of

D

. If

f(z)=\sum_{n=0}^{\infty} a_nz^n

, then we shall denote

S_n(f,z)=\sum_{k=0}^n a_kz^k

. A function

f\in H(D)

is called universal if, for every continuous function

g\colon\partial D\rightarrow \mathbb{C}

and for every

\varepsilon >0

, there are partial sums

S_{n(j)}(f,z)

approximating

g

uniformly on the arc

\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}

. Prove that the set of universal functions contains a dense

G_{\delta}

subset of

H(D)

Problem Statement