Proof of The One-Dimensional Fejer Kernel: Regularity, Nonnegativity, Mass and Far-Field Decay

Each result cited is universally quantified over the data in its own statement. Fix $N\in\mathbb{N}$ and write, for $m\in[N]$,

$$c_{m}=\frac{2(N-m)}{N},$$

so that, by the homogeneity of finite sums, claim 3 of Properties of Finite Sums,

$$F_{N}(t)=C_{0}(t)+\sum_{m=1}^{N}c_{m}C_{m}(t)\qquad(t\in\mathbb{R}),$$

the constant term being $C_{0}$ because $C_{0}(t)=1$ for every $t$ by Cell Integrals of the Trigonometric Monomials (i).

Claim 1. By Cell Integrals of the Trigonometric Monomials (i) each $C_{m}$ is continuous on $\mathbb{R}$. Let $P$ be the set of $p\in\mathbb{N}$ for which the map $t\mapsto\sum_{m=1}^{p}c_{m}C_{m}(t)$ is continuous on $\mathbb{R}$, where $c_{m}$ is extended to every $m\in\mathbb{N}$ by the same formula. A sum with one summand equals that summand, by claim 1 of Properties of Finite Sums, so the map for $p=1$ is $c_{1}C_{1}$, continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, which gives scalar multiples; hence $1\in P$. If $p\in P$, the recursion in claim 1 of Properties of Finite Sums writes the map for $p+1$ as the sum of the map for $p$ and $c_{p+1}C_{p+1}$, continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, which gives sums and scalar multiples; hence $p+1\in P$. By Principle of Induction for the Natural Numbers, $P=\mathbb{N}$, so the map $t\mapsto\sum_{m=1}^{N}c_{m}C_{m}(t)$ is continuous on $\mathbb{R}$.

The constant map $C_{0}$ is continuous at every point of $\mathbb{R}$ by claim 1 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, the point being arbitrary, hence continuous on $\mathbb{R}$; so $F_{N}$ is continuous on $\mathbb{R}$ by claim 5 of that theorem. Consequently $F_{N}|_{J}$ is $\lambda_{J}$-integrable by Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function (ii).

For periodicity, let $t\in\mathbb{R}$ and $j\in\mathbb{Z}$, and let $m\in[N]$. The natural number $m$ is an integer, so $mj\in\mathbb{Z}$ by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers; since $2\pi m(t+j)=2\pi mt+2\pi(mj)$, Quarter-Turn Identities and Periodicity of Sine and Cosine (iv) gives $C_{m}(t+j)=C_{m}(t)$. As this holds for every $m\in[N]$, and $C_{0}$ is constant, the displayed formula for $F_{N}$ gives $F_{N}(t+j)=F_{N}(t)$.

Claim 2. Multiplying the defining formula for $F_{N}(t)$ by $N$ and using $N\tfrac{1}{N}=1$ gives

$$N\,F_{N}(t)=N+2\sum_{m=1}^{N}(N-m)\cos(2\pi mt).$$

Multiplying by $(\sin(\pi t))^{2}$ and applying The Dirichlet and Fejer Kernel Identities (ii) gives the assertion.

Claim 3. First suppose $\sin(\pi t)\ne0$. Then $(\sin(\pi t))^{2}$ is nonzero by claim 3 of Zero Products and Elementary Identities in a Field and nonnegative by claim 2 of Nonnegativity of Squares in an Ordered Field, hence positive. Also $(\sin(N\pi t))^{2}$ is nonnegative, by that same claim 2. If $N\,F_{N}(t)$ were negative, then $N\,F_{N}(t)(\sin(\pi t))^{2}$ would be negative, by claim 5 of Elementary Order Arithmetic in an Ordered Field applied to the positive numbers $-N\,F_{N}(t)$ and $(\sin(\pi t))^{2}$ together with claim 2 of Zero Products and Elementary Identities in a Field, contradicting claim 2 of the present lemma. Hence $0\le N\,F_{N}(t)$, and multiplying by the nonnegative $\tfrac{1}{N}$, by claim 5 of Elementary Arithmetic in an Ordered Field together with claim 1 of Zero Products and Elementary Identities in a Field, gives $0\le F_{N}(t)$.

Now suppose $\sin(\pi t)=0$, and let $u\in\mathbb{R}$ satisfy $0<u<1$. By Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine (iii),

$$\sin\bigl(\pi(t+u)\bigr)=\sin(\pi t)\cos(\pi u)+\cos(\pi t)\sin(\pi u)=\cos(\pi t)\sin(\pi u),$$

using $\pi(t+u)=\pi t+\pi u$ and claim 1 of Zero Products and Elementary Identities in a Field. Here $\cos(\pi t)\ne0$: The Pythagorean Identity for Sine and Cosine (i) gives $(\cos(\pi t))^{2}+(\sin(\pi t))^{2}=1$, and $\sin(\pi t)=0$ in the case under consideration, so $(\sin(\pi t))^{2}=0$ by claim 1 of Zero Products and Elementary Identities in a Field and therefore $(\cos(\pi t))^{2}=1$. This is nonzero because $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, so $\cos(\pi t)\ne0$, again by claim 1 of Zero Products and Elementary Identities in a Field. And $\sin(\pi u)\ne0$: the number $\pi$ is positive, since $\pi=2x_{0}$ with $0<x_{0}$ by The Number Pi (i) and The Least Positive Zero of the Cosine (i), while $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so that claim 5 of that lemma gives $0<\pi$; hence $0<\pi u$ and $\pi u<\pi$ by claim 10 of Elementary Order Arithmetic in an Ordered Field applied to $0<u$ and to $u<1$, whence $0<\sin(\pi u)$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (v). So $\sin(\pi(t+u))\ne0$ by claim 3 of Zero Products and Elementary Identities in a Field, and the case already treated gives $0\le F_{N}(t+u)$.

Let $u_{j}=\tfrac{1}{2j}$ for $j\in\mathbb{N}$. Each $u_{j}$ is positive, and $2\le 2j$, because $1\le j$ by claim 4 of Properties of the Order on the Natural Numbers and multiplication on the left by the nonnegative $2$ preserves the order, by claim 5 of Elementary Arithmetic in an Ordered Field; so $u_{j}\le\tfrac{1}{2}$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares (i) and hence $u_{j}<1$, since $\tfrac{1}{2}<1$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Therefore $0\le F_{N}(t+u_{j})$ for every $j\in\mathbb{N}$, by the previous paragraph.

The sequence $(\tfrac{1}{j})_{j\in\mathbb{N}}$ converges to $0$ by claim 3 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, taken with the exponent $1$, its claim 1 identifying the real power with the natural power; hence $(u_{j})_{j\in\mathbb{N}}$ converges to $0$ by Arithmetic of Limits of Real Sequences, being the multiple of that sequence by $\tfrac{1}{2}$. The constant sequence with value $t$ converges to $t$ by Constant Sequences and Index-Shifted Sequences of Real Numbers (i), so $(t+u_{j})_{j\in\mathbb{N}}$ converges to $t$ by Arithmetic of Limits of Real Sequences. Since $F_{N}$ is continuous at $t$ by claim 1, Continuity Between Metric Spaces is Equivalent to Sequential Continuity (i) gives that $(F_{N}(t+u_{j}))_{j\in\mathbb{N}}$ converges to $F_{N}(t)$; and every term of that sequence is nonnegative, so $0\le F_{N}(t)$ by Order Properties of Limits of Real Sequences.

Claim 4. Each restriction $C_{m}|_{J}$ for $m\in[N]$ is $\lambda_{J}$-integrable by Cell Integrals of the Trigonometric Monomials (i), so Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (i) and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (ii), applied on the measure space $(J,\mathcal{B}_{J},\lambda_{J})$ to these restrictions with the coefficients $c_{m}$, give that the restriction of $t\mapsto\sum_{m=1}^{N}c_{m}C_{m}(t)$ to $J$ is integrable with

$$\int_{J}\sum_{m=1}^{N}c_{m}C_{m}\,d\lambda_{J}=\sum_{m=1}^{N}c_{m}\int_{J}C_{m}\,d\lambda_{J}.$$

For $m\in[N]$ one has $1\le m$ by claim 4 of Properties of the Order on the Natural Numbers, so $m\ne0$ because $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field; hence $\int_{J}C_{m}\,d\lambda_{J}=0$ by Cell Integrals of the Trigonometric Monomials (ii). Every summand on the right is therefore $0$ by claim 1 of Zero Products and Elementary Identities in a Field, and the sum is $0$ by claim 3 of Properties of Finite Sums applied with the scalar $0$.

The same clause Cell Integrals of the Trigonometric Monomials (ii), taken with $m=0$, gives $\int_{J}C_{0}\,d\lambda_{J}=1$. Since $F_{N}=C_{0}+\sum_{m=1}^{N}c_{m}C_{m}$ and both restrictions to $J$ are integrable, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$\int_{J}F_{N}\,d\lambda_{J}=\int_{J}C_{0}\,d\lambda_{J}+\int_{J}\sum_{m=1}^{N}c_{m}C_{m}\,d\lambda_{J}=1+0=1 .$$

Claim 5. By Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine (iv), taken with $x=\pi t$ and using $2(\pi t)=2\pi t$, one has $\cos(2\pi t)=1-2(\sin(\pi t))^{2}$, so that

$$\bigl(\sin(\pi t)\bigr)^{2}=\frac{1-C_{1}(t)}{2}\qquad(t\in\mathbb{R}),$$

the number $2$ being positive, hence invertible, by claim 8 of Elementary Order Arithmetic in an Ordered Field. Write $g$ for the map on the right-hand side; it is continuous on $\mathbb{R}$ by claims 1 and 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, since $C_{1}$ is continuous by Cell Integrals of the Trigonometric Monomials (i), and hence continuous on $K_{\delta}$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map, which gives restrictions.

From $2\delta<1$ and the compatibility of the order with addition one gets $\delta<1-\delta$, so $K_{\delta}$ is the closed interval determined by $\delta$ and $1-\delta$ and is nonempty. By Extreme Value Theorem on a Closed Real Interval there is $t_{\delta}\in K_{\delta}$ with $g(t_{\delta})\le g(t)$ for every $t\in K_{\delta}$; put $c_{\delta}=g(t_{\delta})=(\sin(\pi t_{\delta}))^{2}$. Since $0<\delta\le t_{\delta}$ and $t_{\delta}\le1-\delta<1$, and $\pi$ is positive, claim 10 of Elementary Order Arithmetic in an Ordered Field gives $0<\pi t_{\delta}$ and $\pi t_{\delta}<\pi$; so $0<\sin(\pi t_{\delta})$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (v), and $c_{\delta}$ is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field. Note that $c_{\delta}$ depends only on $\delta$.

Let $N\in\mathbb{N}$ and $t\in K_{\delta}$. By The Pythagorean Identity for Sine and Cosine (ii) one has $|\sin(N\pi t)|\le1$, so

$$\bigl(\sin(N\pi t)\bigr)^{2}=\bigl|\sin(N\pi t)\bigr|^{2}\le1,$$

by claim 1 of Nonnegativity of Squares in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, whose arguments are nonnegative, together with $1\cdot1=1$, an axiom of the field $\mathbb{R}$. The number $N\,F_{N}(t)$ is nonnegative by claim 3 and the positivity of $N$, and $c_{\delta}\le g(t)=(\sin(\pi t))^{2}$; so claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative multiplier $N\,F_{N}(t)$, and claim 2 of the present lemma give

$$N\,F_{N}(t)\,c_{\delta}\le N\,F_{N}(t)\bigl(\sin(\pi t)\bigr)^{2}=\bigl(\sin(N\pi t)\bigr)^{2}\le1 .$$

Finally $N\,c_{\delta}$ is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field, so its inverse is positive by claim 7 of that lemma; multiplying the two ends of the display by that inverse, by claim 5 of Elementary Arithmetic in an Ordered Field, and using associativity and commutativity of multiplication gives $F_{N}(t)\le\tfrac{1}{N\,c_{\delta}}$.