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Iosif Pinelis
Iosif Pinelis
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According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

If $f$ is (say) bounded and locally Lipschitz, then the Hermite expansion of $f$ converges to $f$ uniformly on compact sets. This follows e.g. from Theorem 9.1.6 in the book *Orthogonal Polynomials" by Gábor Szegő (1939), by taking the integral in formula (9.1.17) there by parts, noting that the sine integral function is bounded, and then letting $\delta\downarrow0$ in (9.1.17).

Much more on this is said on pp. 250--251 of theSzegő's book *Orthogonal Polynomials" by Gábor Szegő (1939), starting from "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed.

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

Much more on this is said on pp. 250--251 of the book *Orthogonal Polynomials" by Gábor Szegő (1939), starting from "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed.

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

If $f$ is (say) bounded and locally Lipschitz, then the Hermite expansion of $f$ converges to $f$ uniformly on compact sets. This follows e.g. from Theorem 9.1.6 in the book *Orthogonal Polynomials" by Gábor Szegő (1939), by taking the integral in formula (9.1.17) there by parts, noting that the sine integral function is bounded, and then letting $\delta\downarrow0$ in (9.1.17).

Much more on this is said on pp. 250--251 of Szegő's book, starting from "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed.

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Iosif Pinelis
Iosif Pinelis
  • 127.7k
  • 8
  • 107
  • 229

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

Much more on this is said on pp. 250--251 of the book *Orthogonal Polynomials" by Gábor Szegő (1939), starting from "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed.

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).

Much more on this is said on pp. 250--251 of the book *Orthogonal Polynomials" by Gábor Szegő (1939), starting from "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed.

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Iosif Pinelis
Iosif Pinelis
  • 127.7k
  • 8
  • 107
  • 229

According to the conclusion at the bottom of p. 603 of (say) Uspensky's paper, the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours).