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Uniform Lp bounds for periodic Fourier partial sums

Statement

Assume Countable Choice and let $1 < p < \infty$ . With the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus, write $∥ S_{N} ∥ : = ∥ S_{N} ∥_{L^{p} (T; C) \to L^{p} (T; C)}$ for the operator norm of the $N$ -th Fourier partial sum. Then

$\sup_{N \geq 0} ∥ S_{N} ∥ < \infty .$

Moreover the bound is explicit: with $M : = ∥ C_{p} ∥$ the norm of the $L^{p}$ extension of the conjugate function supplied by The Marcel Riesz conjugate-function theorem on the circle, one has $\sup_{N \geq 0} ∥ S_{N} ∥ \leq 2 + M$ .

Facts & Assumptions

Given: Countable Choice, $1 < p < \infty$ , and the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus: characters $e_{k} (x) = e^{2 π i k x}$ , coefficients $\hat{f} (k) = \int_{0}^{1} f (t) e^{- 2 π i k t} d t$ , partial sums $S_{N} f = \sum_{∣ k ∣ \leq N} \hat{f} (k) e_{k}$ , trigonometric polynomials, and convolution.

[F1]

The torus carries the normalized translation-invariant Haar integral $m$ with $m (T) = 1$ , and $∣ \int_{T} f d m ∣ \leq ∥ f ∥_{1}$ for $f \in L^{1} (T)$ . The one-dimensional torus and its normalized Haar integral Period-one Fourier coefficients, partial sums, and convolution on the torus

[F2]

The conjugate function $C$ is defined on trigonometric polynomials coefficientwise by $\hat{C g} (k) = - i sgn (k) \hat{g} (k)$ ; it is complex-linear, kills constants, and the characters satisfy $e_{a} e_{l} = e_{a + l}$ . Conjugate function on the circle

[F3]

For every $1 < p < \infty$ the operator $C$ extends uniquely to a bounded complex-linear $C_{p}$ on $L^{p} (T; C)$ with $∥ C_{p} ∥ = M < \infty$ , and $C_{p} 1 = 0$ . The Marcel Riesz conjugate-function theorem on the circle

[F4]

For $1 \leq p < \infty$ and $f \in L^{p} (T; C)$ , the Cesaro means $σ_{N} f = \frac{1}{N + 1} \sum_{j = 0}^{N} S_{j} f$ are trigonometric polynomials and $∥ σ_{N} f - f ∥_{p} \to 0$ ; hence the trigonometric polynomials are dense in $L^{p} (T; C)$ . Fejer means converge in L^p for 1 <= p < infinity Cesaro and Abel means of a Fourier series

[F5]

On the finite measure space $T$ one has $∥ f ∥_{1} \leq ∥ f ∥_{p}$ for $1 \leq p < \infty$ . Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$

[F6]

Complex $L^{p}$ carries the norm structure of Complex Holder, Minkowski, and the quotient norm, so the triangle inequality applies to finite sums.

[F7]

A trigonometric polynomial whose Fourier coefficients all vanish is the zero polynomial; equivalently, a finite family of distinct characters is linearly independent. The trigonometric characters are orthonormal in $L^{2}$ of the torus

Proof

technique · direct

1.1F1F3F5F6

Define $P_{+} g : = \frac{1}{2} (g + i C_{p} g) + \frac{1}{2} (\int_{T} g d m) 1$ for $g \in L^{p} (T; C)$ , with $1$ the constant function. Then $P_{+}$ is complex-linear and bounded with $∥ P_{+} g ∥_{p} \leq (1 + \frac{M}{2}) ∥ g ∥_{p}$ , where $M = ∥ C_{p} ∥$ : indeed the triangle inequality of [F6] gives $∥ \frac{1}{2} (g + i C_{p} g) ∥_{p} \leq \frac{1}{2} (1 + M) ∥ g ∥_{p}$ , and $∥ \frac{1}{2} (\int g) 1 ∥_{p} = \frac{1}{2} ∣ \int g ∣ \leq \frac{1}{2} ∥ g ∥_{1} \leq \frac{1}{2} ∥ g ∥_{p}$ by [F1] and [F5].

1.2F1F2

For a trigonometric polynomial $p$ one has $P_{+} p = \sum_{k \geq 0} \hat{p} (k) e_{k}$ . Indeed [F2] gives $\hat{(p + i C p)} (k) = (1 + sgn k) \hat{p} (k)$ , so $\frac{1}{2} (p + i C p)$ has coefficients $\hat{p} (k)$ for $k > 0$ , $\frac{1}{2} \hat{p} (0)$ at $k = 0$ , and $0$ for $k < 0$ ; the constant function $1$ has coefficients $1$ at $k = 0$ and $0$ elsewhere, and $\int_{T} p d m = \hat{p} (0)$ , so adding $\frac{1}{2} \hat{p} (0) 1$ yields exactly the coefficients $\hat{p} (k)$ for $k \geq 0$ and $0$ for $k < 0$ .

1.3F2F6

For $a \in Z$ define the modulation $M_{a} h : = e_{a} h$ , a complex-linear map on $L^{p} (T; C)$ . Since $∣ e_{a} ∣ = 1$ , one has $∣ M_{a} h ∣ = ∣ h ∣$ pointwise and hence $∥ M_{a} h ∥_{p} = ∥ h ∥_{p}$ : each $M_{a}$ is an isometry. For a trigonometric polynomial $h$ and every $k$ , the coefficients satisfy $\hat{M_{a} h} (k) = \hat{h} (k - a)$ , because $e_{a} e_{l} = e_{a + l}$ in [F2] gives $\int e_{a} h e_{- k} d m = \int h e_{- (k - a)} d m$ .

1.4F1F5F6

$S_{N}$ is bounded on $L^{p}$ : for $f \in L^{p}$ each $∣ \hat{f} (k) ∣ \leq ∥ f ∥_{1} \leq ∥ f ∥_{p}$ by [F1] and [F5], so the defining finite sum gives $∥ S_{N} f ∥_{p} \leq \sum_{∣ k ∣ \leq N} ∥ f ∥_{p} = (2 N + 1) ∥ f ∥_{p}$ .

2.1step 1.2step 1.3F7

For every trigonometric polynomial $p$ and every $N \geq 1$ , $S_{N} p = M_{- N} P_{+} M_{N} p - M_{N + 1} P_{+} M_{- (N + 1)} p$ . Indeed, by 1.2 and 1.3 the left-hand side of the identity has coefficients $\hat{(M_{- N} P_{+} M_{N} p)} (k) = \hat{(P_{+} M_{N} p)} (k + N) = 1_{{k + N \geq 0}} \hat{p} (k)$ , and $\hat{(M_{N + 1} P_{+} M_{- (N + 1)} p)} (k) = 1_{{k - (N + 1) \geq 0}} \hat{p} (k)$ ; their difference has coefficients $1_{{∣ k ∣ \leq N}} \hat{p} (k) = \hat{S_{N} p} (k)$ for every $k$ , and two trigonometric polynomials with equal coefficients are equal by [F7].

2.2step 1.1step 1.3F6

The right-hand side of 2.1 defines a bounded operator on $L^{p}$ with norm at most $2 (1 + \frac{M}{2}) = 2 + M$ : by 1.3 each modulation is an isometry and by 1.1 $∥ P_{+} ∥ \leq 1 + \frac{M}{2}$ , so the triangle inequality of [F6] bounds the difference of the two composites by $2 (1 + \frac{M}{2})$ .

3.1step 1.4step 2.1step 2.2F4

The identity of 2.1 holds for every $f \in L^{p} (T; C)$ , not only for trigonometric polynomials: both sides are bounded operators on $L^{p}$ by 1.4 and 2.2, they agree on the set of trigonometric polynomials, and that set is dense in $L^{p}$ by [F4]; two bounded operators agreeing on a dense set agree everywhere.

4.1step 1.4step 2.2step 3.1∎

By 3.1 and 2.2, $∥ S_{N} f ∥_{p} \leq (2 + M) ∥ f ∥_{p}$ for every $f \in L^{p}$ and every $N \geq 1$ , while for $N = 0$ step 1.4 gives $∥ S_{0} f ∥_{p} = ∣ \hat{f} (0) ∣ \leq ∥ f ∥_{p} \leq (2 + M) ∥ f ∥_{p}$ . Hence $∥ S_{N} ∥ \leq 2 + M$ for every $N \geq 0$ and $\sup_{N \geq 0} ∥ S_{N} ∥ \leq 2 + M < \infty$ , which is the assertion.

Depends on

Used by

Periodic Fourier partial sums converge in the strict Lp range Theorem

Dependency tree · two levels

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Sources

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