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It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\, . $$.

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).

It doesn't seem true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset (1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\, . $$

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).

It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\\ . $$.

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).

It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\, . $$.

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).

It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\\ . $$.

It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$

with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\\ . $$.

(edit) I don't know if I was clear enough. It's just this:

1. Any orthonormal family of continuous functions on $[a,b]$ can be the family of eigenfunctions of a compact operator with a kernel satisfying the hypotheses of Mercer's theorem (just choose correspondingly the family of positive eigenvalues decaying to zero fast enough.)

2. There are orthonormal families of continuous functions with unbounded uniform norm (for instance, a sequence of continuous functions with unit $L^2$ norm and disjoint support in $[a,b]$).