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A bounded continuous normalized function that is not of positive type

Statement

On the additive topological group $G = (R, +)$ , let $f (t) = \exp (- t^{4}) .$ This function is real-valued, even, continuous, bounded by $∣ f (t) ∣ \leq 1$ , and normalized by $f (0) = 1$ , but it is not of positive type. In the positive-type matrix for $g_{1} = 0$ , $g_{2} = \frac{1}{2}$ , $g_{3} = 1$ , the coefficient vector $(1, - 2, 1)$ has a negative quadratic form.

Facts & Assumptions

[A1]

Positive type requires the matrix $(φ (g_{i}^{- 1} g_{j}))_{i, j}$ to be positive semidefinite for every finite list and every complex coefficient vector (Continuous positive-type functions and normalization).

[A2]

The real exponential is continuous and strictly increasing (The exponential function is strictly increasing).

[A3]

For every real $x$ , $\exp (x) > 0$ , $\exp (- x) = 1 / \exp (x)$ , and $\exp (0) = 1$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ).

[A4]

For every real $x$ , $1 + x \leq \exp (x)$ ( $1 + x \leq \exp (x)$ for every real $x$ , hence $(1 - p)^{m} \leq \exp (- m p)$ ).

[A5]

Reciprocation reverses strict inequalities between positive reals (Inverses of positives are positive, and reciprocation reverses order).

[A6]

Every polynomial function on $R$ is continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).

[A7]

A composite of continuous real functions is continuous (A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs).

[A8]

The square of every nonzero real number is positive (Squares of nonzero elements are positive).

[A9]

Integer powers are defined by finite repeated multiplication (Integer powers $a^{m}$ ).

[A10]

$1 > 0$ in $R$ (The multiplicative identity is positive).

[A11]

A topological group has continuous multiplication and inversion (Topological group: multiplication and inversion are continuous).

[A12]

The positive reals are closed under addition (Ordered field).

Proof

Given: The additive real group with its usual topology and the function $f (t) = \exp (- t^{4})$ .

Proof technique: direct.

1.1A2A6A7A11algebra

Addition on $R$ is continuous because $∣ (x + y) - (x_{0} + y_{0}) ∣ \leq ∣ x - x_{0} ∣ + ∣ y - y_{0} ∣$ , and inversion $x \mapsto - x$ preserves distances; hence the usual additive group satisfies [A11]. The maps $t \mapsto t^{4}$ and $t \mapsto - t^{4}$ are polynomials and are continuous by [A6]. Composing with the continuous exponential by [A2] and [A7] proves that $f$ is continuous.

1.2A2A3A8A9

Write $t^{4} = (t^{2})^{2}$ . If $t^{2} = 0$ then $t^{4} = 0$ ; otherwise [A8] applied to $t^{2}$ gives $t^{4} > 0$ . Also $(- t)^{4} = t^{4}$ by finite multiplication, so $f$ is even. By [A3] and strict monotonicity in [A2], $0 < f (t) = \exp (- t^{4}) \leq \exp (0) = 1$ for all $t$ , and $f (0) = 1$ . Thus $f$ is real-valued, bounded by $1$ in modulus and normalized at the identity.

2.1A1step 1.2algebra

Put $a = \exp (- 1 / 16)$ and $b = \exp (- 1)$ . The matrix on the listed points is $M = (\begin{array}{ccc} 1 & a & b \\ a & 1 & a \\ b & a & 1 \end{array})$ , since $f$ is even. For $c = (1, - 2, 1)$ , direct multiplication gives $c^{*} M c = 6 - 8 a + 2 b$ .

3.1A1A3A4A5A10A12step 2.1∎

Applying [A4] at $x = - 1 / 16$ gives $a \geq 15 / 16$ , and at $x = 1$ gives $2 \leq \exp (1)$ . By [A10] and [A12], $2 = 1 + 1 > 0$ . By [A3], $b = 1 / \exp (1) > 0$ ; if $\exp (1) = 2$ then $b = 1 / 2$ , while if $2 < \exp (1)$ then [A5] gives $b < 1 / 2$ . Hence $b \leq 1 / 2$ , and step 2.1 yields $c^{*} M c \leq 6 - 8 (15 / 16) + 2 (1 / 2) = - 1 / 2 < 0$ . Thus $M$ is not positive semidefinite by [A1], so $f$ is not of positive type.

Depends on

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Sources

Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition C.4.1 and Proposition C.4.2, Appendix C, printed pp. 373–374 (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, §3.4 (standard reference, not scraped)