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Periodic Fourier partial sums converge in the strict Lp range

Statement

Assume Countable Choice, and use the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus and The one-dimensional torus and its normalized Haar integral: the torus $T = R / Z$ carries normalized Haar measure $m$ with $m (T) = 1$ , and $S_{N} f = \sum_{∣ k ∣ \leq N} \hat{f} (k) e_{k}$ for $f \in L^{p} (T; C)$ .

For every $1 < p < \infty$ and every $f \in L^{p} (T; C)$ , $∥ S_{N} f - f ∥_{p} ⟶ 0 (N \to \infty) .$
At the endpoints the operator norms grow at least as the Lebesgue constants. For every $N \geq 1$ , $∥ S_{N} ∥_{L^{1} \to L^{1}} \geq \int_{T} ∣ D_{N} ∣ d m \geq \frac{1}{3 π} \log (N + 1), ∥ S_{N} ∥_{L^{\infty} \to L^{\infty}} \geq \int_{T} ∣ D_{N} ∣ d m \geq \frac{1}{3 π} \log (N + 1) .$ Hence both families ${(∥ S_{N} ∥_{L^{1} \to L^{1}})}_{N \geq 0}$ and ${(∥ S_{N} ∥_{L^{\infty} \to L^{\infty}})}_{N \geq 0}$ are unbounded, and there exist $f \in L^{1} (T; C)$ and $g \in L^{\infty} (T; C)$ such that $(S_{N} f)$ fails to converge in $L^{1} (T; C)$ and $(S_{N} g)$ fails to converge in $L^{\infty} (T; C)$ .

No failure of weak-type $(1, 1)$ or of any endpoint mapping weaker than norm convergence is asserted.

Facts & Assumptions

Given: Countable Choice, the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus and The one-dimensional torus and its normalized Haar integral, and the $L^{p} (T; C)$ norms of Complex Lp classes and Euclidean test-function conventions.

[F1]

The torus integral is normalized, $m (T) = 1$ , and translation invariant: $\int_{T} h (x - t) d m (x) = \int_{T} h (u) d m (u)$ for integrable $h$ . The character $e_{k} (x) = e^{2 π i k x}$ , the coefficient $\hat{f} (k) = \int_{0}^{1} f (t) e^{- 2 π i k t} d t$ , the partial sum $S_{N} f = \sum_{∣ k ∣ \leq N} \hat{f} (k) e_{k}$ , trigonometric polynomials and torus convolution $(f * g) (x) = \int_{0}^{1} f (x - t) g (t) d t$ are defined as in the cited definition, as is $∥ f ∥_{p}$ for the normalized measure. Period-one Fourier coefficients, partial sums, and convolution on the torus The one-dimensional torus and its normalized Haar integral

[F2]

For every one-period integrable $f$ , every $N \geq 0$ and every $x$ , $S_{N} f (x) = (f * D_{N}) (x) = \int_{0}^{1} f (x - t) D_{N} (t) d t$ . Fourier partial sums are Dirichlet convolutions

[F3]

$D_{N} = 1 + 2 \sum_{k = 1}^{N} \cos (2 π k t)$ is real, even, continuous and bounded, $\int_{T} D_{N} d m = 1$ , and $∥ D_{N} ∥_{1} < \infty$ . The Fejer kernel $F_{M} = \frac{1}{M + 1} \sum_{j = 0}^{M} D_{j}$ satisfies $F_{M} \geq 0$ and $\int_{T} F_{M} d m = 1$ , so $∥ F_{M} ∥_{1} = 1$ . Dirichlet and Fejer kernels The Fejer kernel is a positive approximate identity

[F4]

For $g \in L^{p} (T; C)$ and $1 \leq p < \infty$ the Cesaro means satisfy $σ_{M} g = g * F_{M}$ , are trigonometric polynomials, and $∥ σ_{M} g - g ∥_{p} \to 0$ ; hence trigonometric polynomials are dense in $L^{p} (T; C)$ . For a trigonometric polynomial $P$ one has $S_{N} P = P$ whenever $N \geq \deg P$ . Fejer means converge in L^p for 1 <= p < infinity Cesaro and Abel means of a Fourier series

[F5]

For every $1 < p < \infty$ one has $C_{p} : = \sup_{N \geq 0} ∥ S_{N} ∥_{L^{p} \to L^{p}} < \infty$ . Uniform Lp bounds for periodic Fourier partial sums

[F6]

For every $N \geq 1$ , $∥ S_{N} : C (T) \to C (T) ∥ = \int_{T} ∣ D_{N} ∣ d m \geq \frac{1}{3 π} \log (N + 1)$ . Fourier partial-sum operator norm equals the Lebesgue constant

[F7]

For integrable complex $h$ one has $∣ \int h d m ∣ \leq \int ∣ h ∣ d m$ and $∥ f + g ∥_{p} \leq ∥ f ∥_{p} + ∥ g ∥_{p}$ ; Tonelli's theorem applies to nonnegative measurable functions on the finite product $T \times T$ . The modulus of an integral is bounded by the integral of the modulus Complex Holder, Minkowski, and the quotient norm Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F8]

For every $1 \leq p \leq \infty$ the space $L^{p} (T; C)$ is complete. Complex Lp completeness and almost-everywhere subsequences

[F9]

(Sequential uniform boundedness.) If $X$ is a Banach space, $Y$ a normed space over the same field, and $T_{k} : X \to Y$ , $k \in N$ , are bounded linear maps with $∥ T_{k} x ∥ \leq M_{x}$ for every $k$ and every $x \in X$ , then $\sup_{k} ∥ T_{k} ∥ < \infty$ . Sequential uniform boundedness under countable choice

Proof

technique · direct

1.1F1F2F3F7givenalgebra

For $f \in L^{\infty} (T; C)$ , the convolution formula [F2] and the integral triangle inequality [F7] give $∣ S_{N} f (x) ∣ \leq \int_{T} ∣ f (x - t) ∣ ∣ D_{N} (t) ∣ d m (t) \leq ∥ f ∥_{\infty} ∥ D_{N} ∥_{1}$ for every $x$ , so $S_{N}$ is bounded on $L^{\infty}$ with $∥ S_{N} f ∥_{\infty} \leq ∥ f ∥_{\infty} ∥ D_{N} ∥_{1}$ . For $f \in L^{1} (T; C)$ , Tonelli and the translation invariance of [F1] give $∥ S_{N} f ∥_{1} \leq \int_{T} \int_{T} ∣ f (x - t) ∣ ∣ D_{N} (t) ∣ d m (t) d m (x) = ∥ f ∥_{1} ∥ D_{N} ∥_{1}$ , so $S_{N}$ is bounded on $L^{1}$ . Both bounds are finite by [F3].

1.2F1F2F3F4algebra

For all $M, N \geq 0$ one has $S_{N} (F_{M}) = F_{M} * D_{N} = D_{N} * F_{M} = σ_{M} (D_{N})$ : the first equality is [F2], the second is the substitution $t \mapsto x - t$ in the absolutely convergent torus convolution, and the third is [F4]. Since $∥ F_{M} ∥_{1} = 1$ by [F3], the identity and the Fejer convergence of [F4] give $∥ S_{N} (F_{M}) - D_{N} ∥_{1} = ∥ σ_{M} (D_{N}) - D_{N} ∥_{1} \to 0$ as $M \to \infty$ .

1.3F2F6algebra

For $N \geq 1$ , the unit ball of $C (T)$ is contained in the unit ball of $L^{\infty} (T; C)$ , and for continuous $f$ the partial sum $S_{N} f$ is a trigonometric polynomial, whose essential supremum equals its supremum; hence $∥ S_{N} ∥_{L^{\infty} \to L^{\infty}} \geq ∥ S_{N} : C (T) \to C (T) ∥ = \int_{T} ∣ D_{N} ∣ d m \geq \frac{1}{3 π} \log (N + 1)$ by [F6].

1.4F4F5F7algebra

Let $1 < p < \infty$ , $f \in L^{p} (T; C)$ and $ε > 0$ . By [F4] choose a trigonometric polynomial $P$ with $∥ f - P ∥_{p} < ε / (C_{p} + 2)$ , where $C_{p}$ is the finite bound of [F5]. For $N \geq \deg P$ one has $S_{N} P = P$ by [F4], so the triangle inequality [F7] and the bound [F5] give $∥ S_{N} f - f ∥_{p} \leq ∥ S_{N} (f - P) ∥_{p} + ∥ P - f ∥_{p} \leq (C_{p} + 1) ∥ f - P ∥_{p} < ε$ . Hence $S_{N} f \to f$ in $L^{p}$ for every $1 < p < \infty$ .

2.1step 1.2F3algebra

For fixed $N \geq 0$ and every $M \geq 0$ , step 1.2 and [F3] give $∥ S_{N} (F_{M}) ∥_{1} \leq ∥ S_{N} ∥_{1 \to 1} ∥ F_{M} ∥_{1} = ∥ S_{N} ∥_{1 \to 1}$ , while $∥ S_{N} (F_{M}) ∥_{1} \to ∥ D_{N} ∥_{1}$ ; therefore $∥ S_{N} ∥_{L^{1} \to L^{1}} \geq ∥ D_{N} ∥_{1} = \int_{T} ∣ D_{N} ∣ d m$ .

3.1step 1.3step 2.1algebra

By step 1.3 and step 2.1, for every $N \geq 1$ both endpoint norms satisfy $\max (∥ S_{N} ∥_{L^{1} \to L^{1}}, ∥ S_{N} ∥_{L^{\infty} \to L^{\infty}}) \geq \int_{T} ∣ D_{N} ∣ d m \geq \frac{1}{3 π} \log (N + 1)$ , and $\frac{1}{3 π} \log (N + 1) \to \infty$ . Hence $\sup_{N} ∥ S_{N} ∥_{L^{1} \to L^{1}} = \sup_{N} ∥ S_{N} ∥_{L^{\infty} \to L^{\infty}} = \infty$ , which is the norm-growth assertion of part 2.

4.1step 1.1step 3.1F8F9given

Suppose no $f \in L^{1} (T; C)$ failed to converge. Then each $f$ would have $\sup_{N} ∥ S_{N} f ∥_{1} < \infty$ , and since $L^{1} (T; C)$ is Banach by [F8] and each $S_{N}$ is a bounded linear operator on it by step 1.1, the sequential uniform boundedness principle [F9] would give $\sup_{N} ∥ S_{N} ∥_{L^{1} \to L^{1}} < \infty$ , contradicting step 3.1. Hence there is $f \in L^{1} (T; C)$ with $\sup_{N} ∥ S_{N} f ∥_{1} = \infty$ ; if $(S_{N} f)$ converged to some $g$ in $L^{1}$ , then $∥ S_{N} f ∥_{1} \leq ∥ g ∥_{1} + 1$ for all large $N$ , a contradiction. So $(S_{N} f)$ does not converge in $L^{1}$ .

4.2step 1.1step 3.1F8F9given

The same argument with $L^{\infty} (T; C)$ in place of $L^{1} (T; C)$ : each $S_{N}$ is bounded on $L^{\infty}$ by step 1.1, this space is Banach by [F8], and $\sup_{N} ∥ S_{N} ∥_{L^{\infty} \to L^{\infty}} = \infty$ by step 3.1, so [F9] supplies $g \in L^{\infty} (T; C)$ with $\sup_{N} ∥ S_{N} g ∥_{\infty} = \infty$ , and $(S_{N} g)$ does not converge in $L^{\infty}$ .

5.1step 1.4step 4.1step 4.2step 1.3step 2.1∎

Step 1.4 proves part 1, and steps 4.1 and 4.2 together with the norm lower bounds of steps 1.3 and 2.1 prove part 2.

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Sources

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)